CHAPTER 5Relationships within Triangles

By Zachary Chin and Hyunsoo Kim

THIS is a triangle.

For those of you who are unaware…

How to draw a Triangle 1 2

3Step 1: Draw a line segment.Step 2: Draw another line segment with a common endpoint.Step 3: Connect with another line segment.

NO DUH!

5.1: Midsegments and Coordinate Proofs

MidsegmentsA midsegment is a line in a triangle that connects the midpoints of two sides of a triangle. Every triangle has three midsegments.

Midsegment TheoremThe segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that third side.

>

>

A

B

D

C

E

DE AC and DE=½AC

Example #1Find the length of the midsegment (DE is a midsegment)

A

B CD

E 20

DE = 10

DE = ½AC, so…

Coordinate ProofsA coordinate proof involves placing geometric figures in a coordinate plane. When you use variables to represent the coordinates of a figure in a coordinate proof, the results are true for all figures of that type.

Coordinate ProofsTriangle:

(a, b)

(0, 0) (c, 0)

y

x

Example #2Place a regular octagon in the coordinate plane.

(0, a)

(b, 0)

(c, 0)

(d, a)

(0, e)

(d, e)

(c, f)(b, f)

y

x

5.2: Use Perpendicular Bisectors

A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector.

A point is equidistant from two figures if the point is the same distance from each figure.

Perpendicular Bisector TheoremIn a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

A B

C

P

If CP is the perpendicular bisector of AB, then CA = CB.

Converse of the Perpendicular Bisector Theorem

A B

C

P

D

In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

If DA = DB, then D lies on the perpendicular bisector of AB.

A B

C

P

CP is the perpendicular bisector of AB. Find CB.

Example #3

10x2+15x-6 -6x+4

CB = 1.6 units

Use CA = CB, so…

10x+15x-6 = -6x+4Solve for x, plug it into -6x+4…

When three or more lines, rays, or segments intersect in the same point, they are called concurrent lines, rays, or segments. The point of intersection of the lines, rays, or segments is called the point of concurrency.

Concurrency of Perpendicular Bisectors TheoremThe perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of a triangle.

circumcentercircumscribed circle

5.3: Use Angle Bisectors of Triangles

Angle Bisector Theorem:

If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

Converse of the Angle Bisector Theorem:

If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.

Concurrency of Angle Bisectors of a Triangle

The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

incenter

inscribed circle

Example #4

3x+2

6x-4

Find the value of x.

x = 2

BD = CD, so…

6x-4 = 3x+2

5.4: Use Medians and Altitudes

A median of a triangle is a segment from a vertex to the midpoint of the opposite vertex.

The point of concurrency, called the centroid, is inside the triangle.

Concurrency of Medians of a Triangle

The medians of a triangle intersect at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side.

For example, AG = 2/3 AMA

centroid

An altitude of a triangle is the perpendicular segment from the vertex to the opposite side or to the line that contains the opposite side.

Concurrency of Altitudes of a Triangle

The lines containing the altitudes of a triangle are concurrent.

orthocenter

Example #5

If AG = 2x+3, and AMA = 6x+6, find the value of x.

AG = 2/3 AMA

2x+3 = 2/3 (6x+6) x = -½

5.5 Use Inequalities in a Triangle

Theorem 5.10If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.

8

12

This angle is larger than…

This one

Theorem 5.11

A

BC

If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

AC < BC because m∠A is greater than m∠B.

Example #6Given that m∠A is greater than m∠C, which side is longer, AB or BC?

The answer is BC.

The Triangle InequalityNot all triangles can be made with any lengths. If you have a “triangle” that has sides of 2, 2, and 4, this triangle is not possible to draw. 4=4, as 2+2=4, so that means that this triangle will become a straight line. This “triangle inequality” leads to…

Triangle Inequality TheoremThe sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Example #7Is it possible to make a triangle with sides of 43.6, 57.2, and 101.4?

No, because 100.8 (43.6+57.2) is less than 101.4.

5.6: Inequalities in Two Triangles

Converse of the Hinge Theorem:

If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.

Hinge Theorem:

If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

Example #8

53

Given BC and EF, which is larger, ∠A or ∠D?

∠A

Because 5 > 3.