GeometryChapter 6.1 Medians

Objectives:• Identify medians of triangles

A median is a segment that joins a vertex of the triangle and the midpoint of the opposite side.

A

C BD

Median AD creates two segments, BD and CD.

Is 𝐵𝐷 ≅ 𝐶𝐷? Yes

How many medians does a triangle have?

3

Notice that all the medians intersect a one point, called the centroid.

In Physics, we call this the center of mass.

Find the midpoint of a line using a compass.

GeometryChapter 6.1 Medians

Objectives:• Identify medians of triangles

When three or more lines or segments intersect at the same point, the lines are concurrent.

A

C BD

It can be shown that the length from the vertex to the centroid is twice the length from the centroid to the midpoint.

2x

x

Theorem 6.1: The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint.

Bookwork: page231 problems 8-23.

GeometryChapter 6.2 Altitudes and Perpendicular

BisectorsObjectives:• Identify altitudes and perpendiculars of triangles

An altitude of a triangle is a perpendicular segment from a vertex to the opposite side.

A

C BD

Construct the altitude of a triangle.

AD is an altitude of triangle ABC.

How many altitudes can a triangle have?

3

A perpendicular bisector is a line or segment that bisects a side of a triangle.

E

m

Line m is a perpendicular bisector with point E being the midpoint of CB

Bookwork: page238 problems 8-23

GeometryChapter 6.3 Angle Bisectors of Triangles

Objectives:• Identify angle bisectors of triangles

An angle bisector of a triangle is a segment that bisects an angle of the triangle.

A

C BD

Construct an angle bisector.

1 2

∠1 ≅ ∠2

How many angle bisectors does a triangle have?

3

Notice BD does not have to be congruent to CD

Can they be? Yes

Bookwork: page 242, problems 7-20

GeometryChapter 6.4 Isosceles Triangles

Objectives:• Identify properties of isosceles triangles

Is 𝐴𝐵 ≅ 𝐴𝐶 Yes, all radii of a circle are congruent.

A

BC

Construct the angle bisector of angle A.

Construct the midpoint of BC.

AD is the perpendicular bisector of angle A.

Therefore, ∆𝐴𝐶𝐷 ≅ ∆𝐴𝐵𝐷

GeometryChapter 6.4 Isosceles Triangles

Objectives:• Identify properties of isosceles triangles

In Chapter 5.1 we determined an isosceles triangle has two congruent sides.

From the previous demonstration, an isosceles has two congruent angles.

Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Theorem 6-3: The median from the vertex angle of an isosceles triangle lies on the perpendicular bisector of the base and the angle bisector of the vertex angle.

Converse of the Isosceles Triangle Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Theorem 6-5: A triangle is equilateral if only if it is equiangular.

Bookwork: page249, problems 6-21

GeometryChapter 6.5 Right Triangles

Objectives:• Use tests of congruence for right triangles.

In a right triangle, the side opposite the right angle is the hypotenuse. The two sides that form the right angle are called legs.

If the two legs of a right triangle are congruent to two legs of second right triangle, the triangles are congruent by SAS. This leads to the following theorem.

LL Theorem: If two legs of one right triangle are congruent to the corresponding legs of a second right triangle, then the triangles are congruent.

If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and acute angle of a second triangle, the triangles are congruent by AAS.

HA Theorem: If the hypotenuse and acute angle of one right triangle are congruent to the corresponding hypotenuse and acute angle of a second right triangle, then the triangles are congruent.

GeometryChapter 6.5 Right Triangles

Objectives:• Use tests of congruence for right triangles.

If a leg and an acute angle of a right triangle are congruent to the corresponding leg and acute angle of a second right triangle…

LA Theorem: If one leg and acute angle of one right triangle are congruent to the corresponding leg and acute angle of a second right triangle, then the triangles are congruent.

HL Postulate: If the hypotenuse and leg of one right triangle are congruent to the corresponding hypotenuse and leg of a second right triangle, then the triangles are congruent.

Triangles congruent by ASA Triangles congruent by AAS

Bookwork: page 254, problems 7-21

GeometryChapter 6.6 The Pythagorean Theorem

Objectives:• Define and use the Pythagorean Theorem

a

a

a

a

b

b

b

b

c

c

c

c

The area of the red square is?

𝑎 + 𝑏 𝑎 + 𝑏 = 𝑎2 + 2𝑎𝑏 + 𝑏2

The area of a triangle is?

1

2𝑎𝑏 ∙ 4 = 2𝑎𝑏

The area of all red triangles is?

1

2𝑙𝑤 𝑜𝑟

1

2𝑏ℎ

The area of the blue square is?

𝑎2 + 2𝑎𝑏 + 𝑏2 − 2𝑎𝑏 = 𝑎2 + 𝑏2

𝑎2 + 𝑏2 = 𝑐2

A Square

RIGHT?

Geometry

The Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse, (c), is equal to the sum of the squares of the lengths of the legs (a and b).

Objectives:• Define and use the Pythagorean Theorem

Chapter 6.6 The Pythagorean Theorem

a

b

c 𝑐2 = 𝑎2 + 𝑏2

The Converse of Pythagorean Theorem: If the measure of the longest side of a triangle, c, the measure of the lengths of the other two sides, a and b, and 𝑎2 + 𝑏2 = 𝑐2, then the triangle is a right triangle.

Bookwork: page 260, problems 17 – 40; emphasis on problem 40.

Geometry

Objectives:• Calculate the distance of two points on the coordinate plane

Chapter 6.7 Distance on the Coordinate Plane

A B

CGiven two points, A(-4, -3) and C(4, 4)

Find the distance between A and C.

1. Draw point B(4, -3).

2. Find the distance of AB.

3. Find the distance of BC.

4. What kind of triangle is ∆𝐴𝐵𝐶.

5. What theorem can calculate AC?

6. 𝐴𝐶 = −4 − 4 2 + 4 − (−3) 2

7. AC = 10.630…

Geometry

Bookwork: page 266, problems 12 - 29

Chapter 6.7 Distance on the Coordinate Plane

Objectives:• Calculate the distance of two points on the coordinate plane

From this activity, we can write the following theorem:

The Distance Formula Theorem: the distance between two points on the Cartesian plane with coordinates 𝑥1, 𝑦1 𝑎𝑛𝑑 𝑥2, 𝑦2 is

𝑑 = 𝑥2 − 𝑥1 2 + 𝑦2 − 𝑦1 1

GeometryPythagorean Triples

Objectives:• Discuss what is a Pythagorean Triple

The last section defined the Pythagorean Theorem as 𝑎2 + 𝑏2 = 𝑐2.

A Pythagorean Triple is a group of three whole numbers that satisfies the Pythagorean Theorem, where side c is the hypotenuse.

(3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25)

(20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53)

(11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73)

(13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97)

There are 16 primitive Pythagorean triples with c ≤ 100:

GeometryPythagorean Triples

Objectives:• Discuss what is a Pythagorean Triple

A B C X Y (0, 1) (0, -1) Perpendicular

3 4 5 0.6 0.8 -0.33333 3 -1

5 12 13 0.384615 0.923077 -0.2 5 -1

7 24 25 0.28 0.96 -0.14286 7 -1

8 15 17 0.470588 0.882353 -0.25 4 -1

9 40 41 0.219512 0.97561 -0.11111 9 -1

11 60 61 0.180328 0.983607 -0.09091 11 -1

12 35 37 0.324324 0.945946 -0.16667 6 -1

13 84 85 0.152941 0.988235 -0.07692 13 -1

15 112 113 0.132743 0.99115 -0.06667 15 -1

16 63 65 0.246154 0.969231 -0.125 8 -1

17 144 145 0.117241 0.993103 -0.05882 17 -1

19 180 181 0.104972 0.994475 -0.05263 19 -1

20 21 29 0.689655 0.724138 -0.4 2.5 -1

20 99 101 0.19802 0.980198 -0.1 10 -1

21 220 221 0.095023 0.995475 -0.04762 21 -1

23 264 265 0.086792 0.996226 -0.04348 23 -1

24 143 145 0.165517 0.986207 -0.08333 12 -1

28 45 53 0.528302 0.849057 -0.28571 3.5 -1

28 195 197 0.142132 0.989848 -0.07143 14 -1

32 255 257 0.124514 0.992218 -0.0625 16 -1

33 56 65 0.507692 0.861538 -0.27273 3.666667 -1

36 77 85 0.423529 0.905882 -0.22222 4.5 -1

39 80 89 0.438202 0.898876 -0.23077 4.333333 -1

44 117 125 0.352 0.936 -0.18182 5.5 -1

48 55 73 0.657534 0.753425 -0.375 2.666667 -1

51 140 149 0.342282 0.939597 -0.17647 5.666667 -1

52 165 173 0.300578 0.953757 -0.15385 6.5 -1

57 176 185 0.308108 0.951351 -0.15789 6.333333 -1

60 221 229 0.262009 0.965066 -0.13333 7.5 -1

60 91 109 0.550459 0.834862 -0.3 3.333333 -1

65 72 97 0.670103 0.742268 -0.38462 2.6 -1

68 285 293 0.232082 0.972696 -0.11765 8.5 -1

69 260 269 0.256506 0.966543 -0.13043 7.666667 -1

84 187 205 0.409756 0.912195 -0.21429 4.666667 -1

85 132 157 0.541401 0.840764 -0.29412 3.4 -1

88 105 137 0.642336 0.766423 -0.36364 2.75 -1

95 168 193 0.492228 0.870466 -0.26316 3.8 -1

96 247 265 0.362264 0.932075 -0.1875 5.333333 -1

104 153 185 0.562162 0.827027 -0.30769 3.25 -1

105 208 233 0.450644 0.892704 -0.2381 4.2 -1

115 252 277 0.415162 0.909747 -0.21739 4.6 -1

119 120 169 0.704142 0.710059 -0.41176 2.428571 -1

120 209 241 0.497925 0.86722 -0.26667 3.75 -1

133 156 205 0.64878 0.760976 -0.36842 2.714286 -1

140 171 221 0.633484 0.773756 -0.35714 2.8 -1

160 231 281 0.569395 0.822064 -0.3125 3.2 -1

161 240 289 0.557093 0.83045 -0.30435 3.285714 -1

GeometryPythagorean Triples

Objectives:• Discuss what is a Pythagorean Triple

A B C X Y (0, 1) (0, -1) Perpendicular

3 4 5 0.6 0.8 -0.33333 3 -1

5 12 13 0.384615 0.923077 -0.2 5 -1

7 24 25 0.28 0.96 -0.14286 7 -1

8 15 17 0.470588 0.882353 -0.25 4 -1

9 40 41 0.219512 0.97561 -0.11111 9 -1

11 60 61 0.180328 0.983607 -0.09091 11 -1

12 35 37 0.324324 0.945946 -0.16667 6 -1

13 84 85 0.152941 0.988235 -0.07692 13 -1

15 112 113 0.132743 0.99115 -0.06667 15 -1

16 63 65 0.246154 0.969231 -0.125 8 -1

17 144 145 0.117241 0.993103 -0.05882 17 -1

19 180 181 0.104972 0.994475 -0.05263 19 -1

20 21 29 0.689655 0.724138 -0.4 2.5 -1

20 99 101 0.19802 0.980198 -0.1 10 -1

21 220 221 0.095023 0.995475 -0.04762 21 -1

23 264 265 0.086792 0.996226 -0.04348 23 -1

24 143 145 0.165517 0.986207 -0.08333 12 -1

28 45 53 0.528302 0.849057 -0.28571 3.5 -1

28 195 197 0.142132 0.989848 -0.07143 14 -1

32 255 257 0.124514 0.992218 -0.0625 16 -1

33 56 65 0.507692 0.861538 -0.27273 3.666667 -1

36 77 85 0.423529 0.905882 -0.22222 4.5 -1

39 80 89 0.438202 0.898876 -0.23077 4.333333 -1

44 117 125 0.352 0.936 -0.18182 5.5 -1

48 55 73 0.657534 0.753425 -0.375 2.666667 -1

51 140 149 0.342282 0.939597 -0.17647 5.666667 -1

52 165 173 0.300578 0.953757 -0.15385 6.5 -1

57 176 185 0.308108 0.951351 -0.15789 6.333333 -1

60 221 229 0.262009 0.965066 -0.13333 7.5 -1

60 91 109 0.550459 0.834862 -0.3 3.333333 -1

65 72 97 0.670103 0.742268 -0.38462 2.6 -1

68 285 293 0.232082 0.972696 -0.11765 8.5 -1

69 260 269 0.256506 0.966543 -0.13043 7.666667 -1

84 187 205 0.409756 0.912195 -0.21429 4.666667 -1

85 132 157 0.541401 0.840764 -0.29412 3.4 -1

88 105 137 0.642336 0.766423 -0.36364 2.75 -1

95 168 193 0.492228 0.870466 -0.26316 3.8 -1

96 247 265 0.362264 0.932075 -0.1875 5.333333 -1

104 153 185 0.562162 0.827027 -0.30769 3.25 -1

105 208 233 0.450644 0.892704 -0.2381 4.2 -1

115 252 277 0.415162 0.909747 -0.21739 4.6 -1

119 120 169 0.704142 0.710059 -0.41176 2.428571 -1

120 209 241 0.497925 0.86722 -0.26667 3.75 -1

133 156 205 0.64878 0.760976 -0.36842 2.714286 -1

140 171 221 0.633484 0.773756 -0.35714 2.8 -1

160 231 281 0.569395 0.822064 -0.3125 3.2 -1

161 240 289 0.557093 0.83045 -0.30435 3.285714 -1