…BLM 3–1 Get Readypjk.scripts.mit.edu/lab/2d/BLM_Chap_3_ALL.pdfperpendicular to the chord. 4. a) Verify that the centre of this circle lies on the right bisector of the chord XY

Name: ___________________________________ Date: _______________________________

…BLM 3–1...

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 3–1 Get Ready

Get Ready

Length and Midpoint of a Line Segment 1. Find the midpoint of each line segment.

a) AB with A(−3, 6) and B(5, −2) b) MN with M(−6, 0) and N(4, 8) c) PQ with P(2, 8) and Q(2, −5) d) YZ with Y(−8, 7) and Z(4, 7)

2. Find the exact length of each line segment.

a) CD with C(−3, 5) and D(1, 8) b) GH with G(−4, −3) and H(2, 1) c) KL with K(3, 5) and L(3, −2) d) RS with R(1, −3) and S(5, −3)

Intersection of Lines 3. Find the point of intersection of the lines

represented by each pair of equations. a) y = x + 2 and y = −x − 4 b) y = x − 5 and y = −x + 2 c) y = 3x − 4 and y = x + 2

4. Find the point of intersection of the lines

represented by each pair of equations. a) 3x + y = 6 and 2x − y = 9 b) 2x + y = 6 and x − 2y = 8 c) 3x + 4y = 6 and 4x − 3y = 8

Sum of the Angles in a Triangle 5. Find the measure of the third angle in each

triangle. a) b)

6. Find the measure of the unknown angle in each triangle. a) b)

Types of Quadrilaterals 7. List the key properties of

a) a square b) a rhombus c) a kite

8. Draw an example of each shape and mark all

equal sides and all parallel sides. a) a rectangle b) a trapezoid

Name: ___________________________________ Date: _______________________________

…BLM 3–3...

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 3–3 Section 3.1 Practice Master

Section 3.1 Practice Master

1. The area of DEF is 28 square units. Find the area of DEG.

2. The area of JKM is 16 square units. Find

the area of JKL.

3. a) Draw any isosceles triangle. Then, construct

the median from the vertex between the equal sides. b) List three properties of this median that

a median of a scalene triangle does not have.

c) Verify these properties by measuring your drawing.

4. a) Draw any equilateral triangle. Then,

construct the angle bisector of each vertex. b) Verify that the angle bisectors and the right

bisectors of the sides all meet at the same point.

5. List at least three properties of isosceles

triangles. Explain how you know that every isosceles triangle has each of these properties.

6. Use Technology a) Construct any PQR. Then, construct the

midpoint X of side PQ and the midpoint Y of side QR.

b) Show that line segment XY is parallel to line segment PR.

c) Show that line segment XY is half the length of line segment PR.

7. In this triangle, the right bisectors of the three

sides meet at a single point, called the circumcentre. Investigate whether the right bisectors intersect at a single point in all triangles. Outline your method and describe your findings.

8. a) Investigate whether every triangle has an

orthocentre, a point where the altitudes of all the sides meet. Describe your method and your results. b) Where is the orthocentre located if the

triangle is a right triangle? c) Where is the orthocentre located if the

triangle is an obtuse triangle?

Name: ___________________________________ Date: _______________________________

…BLM 3–4...



1. Determine an equation for the line shown with each triangle.

a) b)

2. a) Verify that AB and MN are parallel.

b) Verify that MN is half the length of AB.

3. a) Verify that DEF is isosceles.

b) Verify that the median from vertex D

is also an altitude of the triangle. 4. Use Technology Use geometry software to

verify your answers to question 3. 5. a) Draw the triangle with vertices D(−2, −5),

E(2, 3), and F(4, −3). b) Find the lengths of the sides of DEF. c) Find the slopes of the sides of DEF. d) Classify DEF. Explain your reasoning.

6. Use Technology Use geometry software to

verify your answers to question 5. 7. a) Draw the triangle with vertices A(−14, 6),

B(2, 0), and C(−10, −6). b) Determine the coordinates of D, the

midpoint of AB, and E, the midpoint of AC. c) Verify that DE is parallel to BC. d) Verify that BC is twice the length of DE.

Name: ___________________________________ Date: _______________________________

…BLM 3–6...



1. a) Which line segments inside parallelogram ABCD are equal in length?

b) Which line segments inside parallelogram

EFGH are equal in length?

2. a) Which line segments inside quadrilateral

PQRS are parallel? b) Which line segments inside quadrilateral

PQRS are equal in length?

3. Which line segments inside quadrilateral JKLM are parallel?

4. Fold a drawing of a rectangle to investigate

the properties of its diagonals. Describe your findings.

5. Verify that the diagonals of a square bisect

each other at right angles and are equal in length. Explain your reasoning.

6. a) Verify that the diagonals of a parallelogram

bisect each other. Explain your reasoning. b) Verify that the diagonals of a rectangle

are equal in length. Explain your reasoning. c) Verify that the diagonals of a kite are

perpendicular and that one of the diagonals bisects the other diagonal. Explain your reasoning.

Name: ___________________________________ Date: _______________________________

…BLM 3–8...



1. Verify that quadrilateral WXYZ is a trapezoid.

2. Verify that quadrilateral PQRS is a rhombus.

3. Verify that quadrilateral DEFG is a kite.

4. a) Draw the quadrilateral with vertices A(−2, 3), B(−5, 1), C(−1, −5), and D(2, −3). b) Verify that quadrilateral ABCD is

a rectangle. c) Verify that the diagonals of ABCD are

equal in length and bisect each other. 5. a) Draw the quadrilateral with vertices

S(−2, 4), T(− 4, −2), U(2, − 4), and V(4, 0). b) Find the midpoint D of side ST, the

midpoint E of side TU, the midpoint F of side UV, and the midpoint G of side VS. Join the midpoints of adjacent sides to form a new quadrilateral DEFG.

c) Verify that opposite sides of DEFG are parallel.

d) Verify that opposite sides of DEFG are equal in length.


answer question 5. Outline your method. 7. a) Draw the trapezoid with vertices P(−3, 3),

Q(2, 4), R(6, −1), and S(− 4, −3). b) Verify that the line segment joining the

midpoints of the non-parallel sides of the trapezoid is parallel to the other two sides.


answer question 7. Outline your method.

Name: ___________________________________ Date: _______________________________

…BLM 3–9...



1. a) Find the coordinates of the midpoint, M, of DE. b) Find the slope of chord DE. c) Verify that OM is perpendicular to DE.

2. a) Verify that the points K(−2, 4), L(4, 4),

and M(3, −1) are equidistant from the point C(1, 2). b) Draw the circle that passes through the

points K, L, and M. 3. a) Explain how you know that the origin is

the centre of the circle represented by x2 + y2 = 34. b) Verify that the points A(−3, 5) and

B(−5, −3) lie on the circle. c) Verify that the line through the origin and

the midpoint of the chord AB is perpendicular to the chord.

4. a) Verify that the centre of this circle lies on the right bisector of the chord XY.

b) Find the radius of the circle.

5. Find the centre of the circle that passes

through the points D(−5, 6), E(−2, 7), and F(2, 5).


answer question 5. Outline your method. 7. For this circle, the diameter is AB and a point

on the circle is C. Verify that ∠ACB = 90°. Explain your reasoning.

Name: ___________________________________ Date: _______________________________

…BLM 3–10... (page 1)

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 3–10 Chapter 3 Review

Chapter 3 Review

3.1 Investigate Properties of Triangles 1. a) Define an altitude.

b) List two additional properties of the altitudes of a triangle.

c) Outline how you could use geometry software to show that the altitudes of all triangles have these properties. 2. a) Show that ABC with vertices A(−3, 3),

B(−1, 5), and C(1, 3) is isosceles.

b) Find the midpoint, M, of side AB and the

midpoint, N, of side BC. c) Verify that the lengths of the medians of

the two equal sides of the isosceles triangle are equal.

3.2 Verify Properties of Triangles 3. a) Verify that PQR is a right triangle.

b) Describe another method that you could

use to verify that PQR is a right triangle.

4. a) Classify XYZ. Explain your reasoning.

b) Verify that the altitude from vertex X

bisects side YZ in XYZ. 5. A triangle has vertices K(–2, 2), L(1, 5), and

M(3, –3). Verify that a) the triangle has a right angle b) the midpoint of the hypotenuse is

equidistant from each vertex

3.3 Investigate Properties of Quadrilaterals 6. a) Verify that the diagonals of a square are

equal and bisect each other at right angles. Explain your reasoning. b) Verify that the diagonals of a

parallelogram bisect each other. Explain your reasoning.

c) Verify that the diagonals of a kite meet each other at right angles and one diagonal bisects the other diagonal. Explain your reasoning. 7. a) Draw any parallelogram ABCD.

b) Show that ABC ≅ CDA. Explain your reasoning. 8. Use Technology Use geometry software to

verify your answer to question 7.

Name: ___________________________________ Date: _______________________________

…BLM 3–10... (page 2)

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 3–10 Chapter 3 Review

3.4 Verify Properties of Quadrilaterals 9. Verify that quadrilateral DEFG is a

parallelogram.

10. Verify that the quadrilateral with vertices

P(2, 3), Q(5, –1), R(10, –1), and S(7, 3) is a rhombus.

11. a) Draw the quadrilateral with vertices

P(−3, 4), Q(−2, 6), R(0, 5), and S(1, −3). b) Classify quadrilateral PQRS. Justify this

classification. c) Verify a property of the diagonals of

quadrilateral PQRS. 12. A quadrilateral has vertices K(–1, 4), L(2, 2),

M(0, –1), and N(–3, 1). Verify that a) KLMN is a square b) each diagonal of KLMN is the

perpendicular bisector of the other diagonal

c) the diagonals of KLMN are equal in length

3.5 Properties of Circles 13. a) Show that X(− 4, −3) and Y(4, 3) are

endpoints of a diameter of the circle x2 + y2 = 25. b) State the coordinates of another point Z,

with integer coordinates, on the circle x2 + y2 = 25.

c) Show that XYZ is a right triangle. 14. a) Verify that the points D(−2, 5) and

E(5, −2) lie on the circle with equation x2 + y2 = 29. b) Verify that the right bisector of the chord

DE passes through the centre of the circle. 15. Find the centre of the circle that passes

through the points P(–9, 5), Q(1, 5), and R(–2, –2).


answer question 15. Outline your method.

Name: ___________________________________ Date: _______________________________

…BLM 3–12...

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 3–12 Chapter 3 Practice Test

Chapter 3 Practice Test

1. An isosceles triangle has A three medians that are all different in

length B three medians that are all equal in length C three medians with two of the medians

equal in length D three medians with two of the medians

different in length 2. The diagonals of a square

A are equal in length and bisect each other at right angles

B are equal in length and bisect each other, but not at right angles

C are different in length and bisect each other at right angles

D are different in length and bisect each other, but not at right angles 3. The three altitudes of a triangle intersect at

a point called the A centroid B incentre C circumcentre D orthocentre

4. A parallelogram is a quadrilateral

A with four equal sides and four 90° angles B that has exactly one pair of parallel sides C with opposite sides parallel and equal D that has no parallel sides

5. The point (4, −1) is on the circle represented

by the equation A x2 + y2 = 3 B x2 + y2 = 15 C x2 + y2 = 17 D x2 + y2 = 29

6. Sketch an example of each quadrilateral.

Show the diagonals on each sketch and indicate whether they are equal in length and whether they bisect each other. a) rectangle b) trapezoid

7. a) Verify that DEF is isosceles. b) Verify that the point M(2, 3) lies on the

altitude of DEF from vertex D.

8. a) Use analytic geometry to verify that

quadrilateral JKLM is a parallelogram. b) Describe how to use geometry software

to answer part a).


W(− 4, −1), X(0, 5), Y(3, 3), and Z(−1, −3). b) Verify that quadrilateral WXYZ is

a rectangle. c) Verify that the diagonals of quadrilateral

WXYZ bisect each other. 10. a) Verify that C(−1, 1) is the centre of the

circle that passes through the points D(2, 5), E(3, −2), and F(2, −3). b) Find the radius of the circle.

Name: ___________________________________ Date: _______________________________

…BLM 3–13...

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited BLM 3–13 Chapter 3 Test

Chapter 3 Test

1. An equilateral triangle has A three medians that are all different in

length B three medians that are all equal in length C three medians with two of the medians

equal in length D three medians with two of the medians

different in length 2. The diagonals of a kite

A are equal in length and bisect each other at right angles

B are equal in length and bisect each other, but not at right angles

C are different in length and intersect each other at right angles

D are different in length and bisect each other, but not at right angles 3. The three medians of a triangle intersect at

a point called the A centroid B incentre C circumcentre D orthocentre

4. A trapezoid is a quadrilateral

A with four equal sides and four 90° angles B that has exactly one pair of parallel sides C with opposite sides parallel and equal D that has no parallel sides

5. The point (5, −2) is on the circle represented

by the equation A x2 + y2 = 3 B x2 + y2 = 21 C x2 + y2 = 23 D x2 + y2 = 29

6. Sketch an example of each quadrilateral.

Show the diagonals on each sketch and indicate whether they are equal in length and whether they bisect each other. a) parallelogram b) rhombus

7. a) Verify that PQR is a right triangle. b) Verify that the point A(1, 4) lies on the

line containing the altitude of PQR drawn from vertex Q.

8. a) Use analytic geometry to verify that

quadrilateral JKLM is a rhombus. b) Describe how to use geometry software to

answer part a).


A(− 4, 3), B(−2, 2), C(−2, −3), and D(−5, 1). b) Verify that quadrilateral ABCD is a kite. c) Verify that the diagonals of quadrilateral

ABCD intersect at right angles. 10. a) Show that D(−24, 7) and M(24, −7) are

endpoints of a diameter of the circle defined by x2 + y2 = 625. b) State the coordinates of another point, C,

on the circle. c) Show that DMC is a right triangle.

…BLM 3–18... (page 1)

Principles of Mathematics 10: Teacher’s Resource Copyright © 2007 McGraw-Hill Ryerson Limited Chapter 3 Practice Masters Answers

BLM Answers

Get Ready

1. a) (1, 2) b) (−1, 4) c) ( )32, 2 d) (−2, 7)

2. a) 5 b) 2 13 c) 7 d) 4

3. a) (−3, −1) b) ( )7 3,2 2

− c) (3, 5)

4. a) (3, −3) b) (4, −2) c) (2, 0) 5. a) ∠B = 58 b) ∠F = 12° 6. a) x = 64 b) y = 84° 7. Answers may vary. For example: a) A square is a quadrilateral with four equal sides

and four 90° angles. b) A rhombus has four equal sides and the opposite

sides are parallel. c) A kite has two pairs of adjacent sides equal, the

diagonals are not equal in length, and the diagonals intersect at right angles. The diagonal between the equal sides is the right bisector of the other diagonal.

8. Diagrams may vary. For example: a) b)

Section 3.1 Practice Master 1. 14 square units 2. 32 square units 3. a) Diagrams may vary. For example:

b) For an isosceles triangle, the median between the

equal sides is also the altitude from that vertex. The median bisects the angle at that vertex and bisects the opposite side. It is the same as the right bisector of the opposite side.

c) Answers will vary.

4. a) Diagrams may vary. For example:

b) Answers may vary. For example: Construct GHI.

Then, construct all three angle bisectors and all three right bisectors and verify that they coincide.

5. Isosceles triangles have two equal sides and two equal angles. The median drawn from the vertex between the equal sides is the same as the angle bisector of the same vertex, and the altitude drawn from the vertex between the equal sides is the same as the right bisector of the opposite side.

6. a) Answers will vary. b) The slopes are equal, so the line segments

are parallel. c) Answers may vary. For example: Measure the

length of each segment to show that XY is half the length of PR.

7. Yes. Explanations may vary. For example: Right bisectors drawn in examples of each type of triangle intersect at a single point in each triangle.

8. a) Yes. Explanations may vary. For example: Altitudes drawn in examples of each type of triangle intersect at a single point.

b) If the triangle is a right triangle, the orthocentre is located at the right angle.

c) If the triangle is an obtuse triangle, the orthocentre is located outside the triangle.


1. a) y = 17 x + 10

7 b) y = 16 x + 3

2

2. a) mAB = 12 and mMN = 1

2 . Since mAB = mMN,

AB is parallel to MN.

b) MN = 3 52

and AB = 3 5 , so MN is half the

length of AB.

…BLM 3–18... (page 2)


3. a) DE = 4 5 , DF = 4 5 , EF = 5 6 Since DE = DF = 4 5 , DEF is isosceles.

b) The midpoint of EF is M(4, 0). Therefore, the median from vertex D is DM.

Since mDM = 13

− and mEF = 3, mDM × mEF = −1, so

DM is perpendicular to EF, which makes DM an altitude of the triangle.

4. Answers may vary. For example: a) Construct DEF and measure the sides to verify

that DE = DF. b) Construct the midpoint, M, of EF. Construct

segment DM, the median from vertex D. Verify that DM is perpendicular to EF, so that it is also the altitude from vertex D.

5. a)

b) DE = 4 5 , EF = 2 10 , DF = 2 10

c) mDE = 2, mEF = 13

, mDF = −3

Since mEF × mDF = −1, ∠DFE = 90°. d) Since EF = DF and ∠DFE = 90°, DEF is an

isosceles right triangle. 6. Answers may vary. For example: a) Construct DEF. b) Measure the lengths of the sides of DEF. c) Measure ∠DFE. d) Since EF = DF and ∠DFE = 90°, DEF is an

isosceles right triangle. 7. a) b) D(−6, 3), E(−12, 0)

c) Since mDE = mBC = 12

, DE is parallel to BC.

d) BC = 6 5 and DE = 3 5 . Therefore, BC is twice the length of DE.

Section 3.3 Practice Master 1. a) AE = CE, BE = DE b) EM = GM, FM = HM 2. a) AB is parallel to DC; AD is parallel to BC. b) AB = DC, AD = BC 3. a) MN is parallel to QR; MR is parallel to NQ. b) MN = QR, MR = NQ 4. Answers may vary. For example: The diagonals of a

rectangle are equal in length and bisect one another. 5. Answers may vary. For example: Construct a square.

Construct the diagonals and their point of intersection. Measure the angles at the point of intersection to verify that the diagonals are perpendicular. Construct the midpoint of each diagonal and verify that it coincides with the point of intersection of the diagonals. Verify that your conclusions do not change as you drag the vertices of the square.

6. a) Answers may vary. For example: Construct a rectangle and its diagonals. Construct the midpoint of each diagonal and verify that it coincides with the point of intersection of the diagonals. Drag the vertices of the rectangle to verify your conclusions.

b) Answers may vary. For example: Construct a rectangle and its diagonals. Measure the diagonals to verify that they are equal in length. Drag the vertices of the rectangle to verify your conclusions.

c) Answers may vary. For example: Construct a kite and its diagonals. Measure the angle at the point of intersection of the diagonals to see that it is 90°. Construct the midpoint of the shorter diagonal and verify that it coincides with the point of intersection of the diagonals. Drag the vertices of the kite to verify your conclusions.


1. mWX = 12

− , mZY = 12

− , WX = 5 , ZY = 3 5 ,

mWZ = 35

, mXY = 5

Since mWX = mZY = 12

− , WX ≠ ZY, and mWZ ≠ mXY,

quadrilateral WXYZ is a trapezoid.

2. PQ = 5, QR = 5, RS = 5, SP = 5, mSP = 43

− ,

mQR = 43

− , mPQ = 0, mRS = 0

Since PQ = QR = RS = SP = 5, mSP = mQR, and mPQ = mRS, quadrilateral PQRS is a rhombus.

…BLM 3–18... (page 3)


3. Since DE = EF = 10 and DG = GF = 5, quadrilateral DEFG is a kite.

4. a)

b) mAB = 23

, mCD = 23

, mBC = 32

− , mAD = 32

− .

Since mAB × mBC = −1, ∠ABC = 90°. Since mBC × mCD = −1, ∠BCD = 90°. Since mCD × mAD = −1, ∠CDA = 90°. Since mAD × mAB = −1, ∠DAC = 90°. Since mAB = mCD, AB is parallel to CD. Since mBC = mAD, BC is parallel to AD. AB = 13 , BC = 2 13 , CD = 13 , AD = 2 13 ;

AB = CD and BC = DA. Since it has four 90° angles and opposite sides are

equal in length, quadrilateral ABCD is a rectangle. c) AC = 65 , BD = 65 The diagonals are equal in length. The midpoint of the diagonal AC and the midpoint

of the diagonal BD are both located at M ( )3 , 12

− − .

Since the midpoints coincide, the diagonals bisect each other.

5. a), b)

c) mDE = −2, mEF = 14

, mFG = −2, mDG = 14

Since mDE = mFG, DE is parallel to FG. Since mEF = mDG, EF is parallel to DG. d) DE = 2 5 , EF = 17 , FG = 2 5 , and

DG = 17 . Since DE = FG and EF = DG, the opposite sides of DEFG are equal in length.

6. Answers may vary. For example: a) Construct quadrilateral STUV. b) Construct the midpoint of each side and display the

coordinates. Construct line segments joining adjacent midpoints.

c) Measure and compare the slopes of the sides of DEFG.

d) Measure and compare the lengths of the sides of DEFG.

7. a)

b) The midpoints of the non-parallel sides PS and QR

are E ( )7 , 02

− and F ( )34,2

, respectively;

mPQ = 15

, mSR = 15 , and mEF = 1

5 .

Since mPQ = mSR = mEF, EF is parallel to PQ and EF is also parallel to SR.

8. Answers may vary. For example: a) Construct trapezoid PQRS. b) Construct the midpoints, E and F, of PS and QR.

Measure the slopes of EF, PQ, and SR. Since the slopes are equal, EF is parallel to both PQ and SR.


1. a) ( )1 7,2 2

− b) mDE = 17

c) mOM = −7. Since mDE × mOM = −1, OM is perpendicular to DE.

2. a) CK = 13 , CL = 13 , CM = 13 Since CK = CL = CM, the points K(−2, 4), L(4, 4),

and M(3, −1) are equidistant from the point C(1, 2). b)

…BLM 3–18... (page 4)


3. a) Answers may vary. For example: The distance formula for any point P(x, y) that is 34 units

from the origin (0, 0) is 2 2( 0) ( 0) 34− + − =x y , which simplifies to x2 + y2 = 34, which is the equation of a circle with centre O(0, 0).

b) Since (−3)2 + 52 = 34 and (−5)2 + (−3)2 = 34, both points lie on the circle.

c) The midpoint of AB is (−4, 1). The slope of chord AB is mAB = 4 and the slope of the right bisector

OM of chord AB is mOM = 14

− .

Since mAB × mOM = −1, OM is perpendicular to AB.

4. a) The midpoint of XY is M ( )3 7,2 2 ; mXY = 37

− ;

the slope of the right bisector of XY is m = 73 ; the

equation of the right bisector of XY is y = 73 x; the

centre of the circle is O(0, 0). Since the point

O(0, 0) lies on the line y = 73 x, the centre of the

circle lies on the right bisector of the chord XY. b) 29 5. (−2, 2) 6. Answers may vary. For example: Construct line

segments DE and EF. Construct the right bisectors of DE and EF. Construct the point of intersection of the right bisectors, and display its coordinates.

7. Answers may vary. For example: Construct line segment CO. Since O is the centre of the circle, it is the midpoint of the diameter AB. Thus, CO is the median from C to AB. CO is also a radius of the circle. Thus, CO = AO = BO. Since CO = AO, AOC is isosceles, and ∠OAC = ∠OCA. Since BO = CO, BOC is isosceles, and ∠OBC = ∠OCB. In ABC,

∠BAC + ∠BCA + ∠CBA = 180° ∠OAC + ∠OCA + ∠OCB + ∠OBC = 180° ∠OCA + ∠OCA + ∠OCB + ∠OCB = 180° 2∠OCA + 2∠OCB = 180° ∠OCA + ∠OCB = 90° ∠ACB = 90°

Chapter 3 Review 1. a) The altitude of a triangle is a line segment from one

vertex of a triangle to the opposite side so that the line segment is perpendicular to the side.

b) The altitudes of a triangle intersect at a point called the orthocentre. If the triangle is a right triangle, the orthocentre is located at the vertex that has the 90° angle.

c) Construct a triangle and its three altitudes. Observe the point of intersection of the altitudes while dragging the vertices of the original triangle. Construct a right triangle and its three altitudes. Note that the orthocentre is located at the vertex that has the 90° angle.

2. a) AB = 2 2 , BC = 2 2 , AC = 4 Since AB = BC, the triangle is isosceles. b) M(−2, 4), N(0, 4) c) CM = 10 , AN = 10 Thus, the medians are equal in length.

3. a) Since mPQ = 23 and mQR = 3

2− , mPQ × mQR = −1

and ∠PQR = 90°. Then, the sides PQ and QR are perpendicular and the triangle is a right triangle.

b) You could also check that the side lengths satisfy the Pythagorean theorem, where the hypotenuse of the triangle is PR.

4. a) XY = 2 13 , XZ = 2 13 , YZ = 2 26 Since XY = XZ, XYZ is isosceles. b) The midpoint of YZ is M(−1, −2), and mYZ = 5.

Since the slope of the line segment from X to

M(−1, −2) is 15

− , the line segment YZ is

perpendicular to XM and is the altitude from X and bisects YZ.

5. a) KL has slope 1 and KM has slope –1. Thus, KL and KM are perpendicular.

b) The midpoint of LM is a distance of 17 from each vertex.

6. a) Answers may vary. For example: Construct a square and its diagonals. Measure the angle at the point of intersection of the diagonals to verify that it is a right angle. Construct the midpoints of the diagonals and verify that they coincide with the point of intersection of the diagonals. Drag the vertices of the square to verify your conclusions.

b) Answers may vary. For example: Construct a parallelogram and its diagonals. Construct the midpoints of the diagonals and verify that they coincide with the point of intersection of the diagonals. Drag the vertices of the parallelogram to verify your conclusions.

c) Answers may vary. For example: Construct a kite and its diagonals. Measure the angle at the point of intersection of the diagonals and verify that its measure is 90°. Construct the midpoint of the shorter diagonal and verify that it coincides with the point of intersection of the diagonals. Drag the vertices of the kite to verify your conclusions.

7. a) Answers will vary. b) In ABC and CDA, since ∠ABC = ∠CDA

(opposite angles in a parallelogram), side AC = side CA (common sides), and

∠BCA = ∠DAC (alternates angles), ABC ≅ CDA (AAS).

…BLM 3–18... (page 5)


8. a) Construct any parallelogram ABCD. b) Construct diagonal AC. Measure the side lengths of

ABC and CDA to verify that the triangles are congruent. Drag the vertices of parallelogram ABCD to verify that the relationship holds true for any parallelogram.

9. mDE = 45 , mFG = 4

5

Since mDE = mFG, DE is parallel to FG.

mEF = 14

− , mDG = 14

−

Since mEF = mDG, EF is parallel to DG. DE = 41 , FG = 41 , EF = 17 , DG = 17 Since DE = FG, EF = DG, and the opposite sides are

parallel, quadrilateral DEFG is a parallelogram. 10. Opposite sides are parallel (two have slope 0 and two

have slope 43

− ) and all sides have length 5, so PQRS

is a rhombus. 11. a) b) Quadrilateral PQRS is a kite. PQ = 5 , QR = 5 , RS = 65 , SP = 65 diagonal PR = 10 , diagonal QR = 90 Since PQ = QR and PS = SP, and the diagonals are

not equal in length, the quadrilateral is a kite. c) Answers may vary. For example: The slope of

diagonal PR is mPR = 13 and the slope of diagonal

QS is mQS = −3. Since mPR × mQS = −1, the diagonals of the kite are

perpendicular to each other.

12. a) Opposite sides are parallel (two have slope 32 and

two have slope 23

− ). Adjacent sides are

perpendicular and all sides have length 13 . Thus, KLMN is a square.

b) The midpoints of the diagonals coincide at

( )1 3,2 2

− , so the diagonals bisect each other. One

diagonal has slope 15 and the other has slope –5.

The diagonals are perpendicular.

13. a) 2 2 25= + =x yL.S. R.S. 2 2( 4) ( 3)

16 925

= − + −= +=

=L.S. R.S.

2 2 25= + =x yL.S. R.S. 2 24 3

16 925

= += +=

=L.S. R.S.

Since the points X(−4, −3) and Y(4, 3) both satisfy the equation, they are on the circle. The radius of the circle x2 + y2 = 25 is r = 5. Since XY = 10, which is double the radius, XY must be a diameter of the circle.

b) Answers may vary. For example: Z(−3, 4) is another point on the circle.

c) XZ = 50 , YZ = 50 , and XY = 10. XZ2 + YZ2 = 2( 50) + 2( 50) = 50 + 50 = 100 = 102 = XY2 These lengths satisfy the Pythagorean theorem with

XY as the hypotenuse, so the triangle is a right triangle.

14. a) Since (−2)2 + (5)2 = 29 and (5)2 + (−2)2 = 29, both points are on the circle.

b) The midpoint of DE is M ( )3 3,2 2

and the slope of

DE is −1. The slope of the perpendicular bisector of DE is m = 1. Therefore, the equation of the perpendicular bisector of DE is y = x, which also passes through the centre of the circle, (0, 0).

15. (–4, 3) 16. Answers may vary. For example: Construct line

segments PQ and QR. Construct the right bisectors of PQ and QR. Construct the point of intersection of the right bisectors, and display its coordinates.

Chapter 3 Practice Test 1. C 2. A 3. D 4. C 5. C

…BLM 3–18... (page 6)


6. a) Diagrams may vary. For example:

b) Diagrams may vary. For example:

7. a) DE = 4 5 , DF = 4 5 , EF = 4 2 Since DE = DF, the triangle is isosceles. b) The midpoint of EF is M(2, 3). Since DEF is

isosceles, the altitude passes through the midpoint of the opposite side, so the point M(2, 3) lies on the altitude of DEF from vertex D.

8. a) JK = 13 , LM = 13 , KL = 26 , JM = 26 ,

mJK = 32 , mLM = 3

2 , mKL = 15

− , mJM = 15−

Since mJK = mLM, JK is parallel to LM. Since mKL = mJM, KL is parallel to JM. Since JK = LM, KL = JM, JK is parallel to LM, and

KL is parallel to JM, quadrilateral JKLM is a parallelogram.

b) Answers may vary. For example: Construct quadrilateral JKLM. Measure the lengths and slopes of the sides to verify that opposite sides are equal and parallel.

9. a)

b) WX = 2 13 , XY = 13 , YZ = 2 13 , WZ = 13

mWX = 32 , mXY = 2

3− , mYZ = 3

2 , mZW = 23

−

Since mWX × mXY = −1, ∠WXY = 90°. Since mXY × mYZ = −1, ∠XYZ = 90°. Since mYZ × mZW = −1, ∠YZW = 90°. Since mZW × mWX = −1, ∠ZWX = 90°. Since WX = YZ and XY = WZ and the triangle has

four 90° angles, the quadrilateral is a rectangle.

c) The midpoint of diagonal WY is M ( )1 , 12

− and the

midpoint of diagonal XZ is N ( )1 , 12

− . Since the

midpoints of the diagonals are the same, the diagonals of quadrilateral WXYZ bisect each other.

10. a) CD = 5, CE = 5, CF = 5 Since CD = CE = CF, C(−1, 1) is the centre of the

circle that passes through the points D(2, 5), E(3, −2), and F(2, −3).

b) 5

Chapter 3 Test 1. B 2. C 3. A 4. B 5. D 6. a) Diagrams may vary. For example:

b) Diagrams may vary. For example:

…BLM 3–18... (page 7)


7. a) mPQ = 2, mQR = 12

− , mQR = 43

−

Since mPQ × mQR = −1, PQR is a right triangle. b) The altitude from Q is contained in the line with

equation 3 13=4 4

+y x . The point A(1, 4) lies on

this line. Thus, A(1, 4) lies on the line containing the altitude.

8. a) JK = 5, KL = 5, LM = 5, JM = 5

mJK = 43

, mKL = 0, mLM = 43

, mJM = 0

Since mJK = mLM, JK is parallel to LM. Since mKL = mJM, KL is parallel to JM. Since the opposite sides are parallel and all four

sides are equal in length, quadrilateral JKLM is a rhombus.

b) Answers may vary. For example: Construct quadrilateral JKLM. Measure the lengths and slopes of the sides. Since all four sides are equal in length and opposite sides are parallel, quadrilateral JKLM is a rhombus.

9. a) b) AB = 5 , AD = 5 , CD = 5, CB = 5

AC = 2 10 , DB = 10 Since AB = AC and CD = CB, and AC ≠ DB, the

quadrilateral is a kite.

c) mAC = −3, mDB = 13

Since mAC × mBD = −1, the diagonals of quadrilateral ABCD intersect at right angles.

10. a) 2 2 625= + =x yL.S. R.S. 2 2( 24) 7

576 49625

= − += +=

=L.S. R.S.

2 2 625= + =x yL.S. R.S. 2 224 ( 7)

576 49625

= + −= +=

=L.S. R.S.

Since the coordinates of D(−24, 7) and M(24, −7) both satisfy the equation, they are on the circle. The radius of the circle x2 + y2 = 625 is r = 25. Since DM = 50, which is double the radius, DM must be a diameter of the circle.

b) Answers may vary. For example: C(7, 24) is another point on the circle.

DC = 25 2 , MC = 25 2 , DM = 50

AC2 + BC2 = ( 25 2 )2 + ( 25 2 )2 = 1250 + 1250 = 2500 = 502 = DM2 These lengths satisfy the Pythagorean theorem with

DM as the hypotenuse, so the triangle is a right triangle.

Documents

…BLM 3–1 Get Readypjk.scripts.mit.edu/lab/2d/BLM_Chap_3_ALL.pdfperpendicular to the chord. 4. a) Verify that the centre of this circle lies on the right bisector of the chord XY