Chapter 5

Relationships in Triangles

5.1 Bisectors, Medians and Altitudes

Special Segments of Triangles

Median – A segment drawn from the vertex of a triangle to the midpoint of the opposite side.Altitude – A segment drawn from the vertex of a triangle perpendicular to the line that contains the opposite side.Angle Bisector – A segment drawn from the vertex of a triangle to the opposite side, dividing the vertex angle into two congruent angles.

Special Segments (Con’t)

Perpendicular Bisector – A segment that starts at the midpoint of a side and is perpendicular to that side.

Notice the similarities and differences among the four different segments.

Medians

A B

CTo draw a median, you start at a vertex

and end at the midpoint of the oppositeside.

Since there are three vertices in a triangle, then there are three medians of a triangle.

DWhat is point D?

E

F

The intersection of the three Medians is called the Centroid.

G

Centroid Theorem – The Centroid is located 2/3rds the way from the vertex to the midpoint on the opposite side.

Angle Bisectors

1 2 A B

C To draw an <bis, you start at a vertex

and end at the opposite side.D D is NOT the midpoint of segment AC.

1

234

5 6

3 4 5 6and E

F

The intersection of the three <bis is called the Incenter.

G

Incenter Theorem – The Incenter is equidistant from the three sides of the triangle.Any point on an <bis is equidistant from the two sides of the angle bisected.

Altitudes

A B

CTo draw an altitude, you start at a vertex

and perpendicular to the line that contains the opposite side.D is NOT the midpoint of segment AB.

D

The intersection of the three altitudes is called the Orthocenter.

Fact: Regardless of what type of triangle you have, the Incenter and Centroid are always in the interior of the triangle. The Orthocenter is interior when the triangle is acute, exterior when the triangle is obtuse and at the right angle when the triangle is a right triangle.

Perpendicular Bisectors

A B

C To draw a perpendicular bisector you start at the midpoint of a side and draw a segment perpendicular at that point.

The intersection of the three perpendicular bisectors is called the Circumcenter.

The circumcenter is equidistant from the three points of the triangle.

Any point on the perpendicular bisector is equidistant from the end points of the segment bisected.The location of the circumcenter is also dependant on theclassification of the triangle, Acute – Interior, Obtuse – ExteriorRight – Midpoint of the Hypotenuse.

5.2 Inequalities and Triangles

Properties of Inequalities

Comparison Property – a<b, a=b or a>b

Transitive Property – If a>b and b>c, then a>cIf a<b and b<c, then a<c

Addition/Subtraction Property – If a<b, then a + c < b + c.

Multiplication/Division Property –If a<b, then ac < bc (unless c is negative)

Exterior Angles

12

3Exterior Angle Theorem – The measure of the exterior is equal to the sum of the tworemote interior angles.(m<1 = m<2 + m<3)

Since the measurement of <1 is equal to the sum of the two remote interior angles, then the measurement of angle 1 must be greater than either remote interior angles. (m<1 > m<2, m<1 > m<3)

Relationships

There is a unique relationship between the sides and the angles that are opposite of the sides.

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

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

Example

A B

C

20

1213

Knowing the measurementsof the three sides we can listthem from largest to smallest.

m<C, m<A, and m<B

Because the relationship of the sides is the same as the relationship of the angles opposite them, we can list the angles opposite them from largest to smallest.

AB, BC, and AC

m<C > m<A > m<B

Another Example

A B

C

100°

55° 25°

Knowing the measurementsof the three angles we can listthem from largest to smallest.

m<C, m<A, and m<BBecause the relationship of the angles is the same as the relationship of the sides opposite them, we can list the sides opposite them from Largest to Smallest.

AB, BC, and AC

AB > BC > AC

Another Example (H)

A C

BD Given the measurements of

the angles find the largest and smallest segment in this picture.

65°60°

55°

70°

50°60°

III

I - <D, <B and <C So; BC > CD > BD

II - <B, <A and <C So; AC > BC > AB

Since BC is in common between the two triangles….

We can conclude that AC is the largest segment and BD is the smallest?

5.3 Indirect Proof

Honor’s Only

Indirect Proof

Another reasoning we can use is Indirect Reasoning – that is prove that your statement can’t be true.

To do an Indirect Proof – you first negate the conclusion.

Then you make that your Hypothesis,

Then prove that the conclusion can’t be reached.

Bottom Line

Given P, prove Q via Indirect Proof. (P → Q)

First, write contra positive (~Q → ~P)

Where ~Q is the given and you want to prove ~P is not possible.

You will not have to do one, just know how to set one up.

5.4 Triangle Inequality

Triangle Inequality

Triangle Inequality Theorem – In a triangle, the sum of the measurements any two sides must be greater than the measurement of the third side.

A C

BAB + BC > ACBC + AC > ABAC + AB > BC

Two Types of Problems

There are two types of problems that deal with the Triangle Inequality Theorem

1st – Given three sides, can you make a triangle?

2nd – Given two sides, what is the range of values possible for the third side

1st Type of Problem

Given three sides, can you make a triangle?

Given segments that measure 3”, 5” and 10”, can you make a triangle?

Short Cut – look at two smallest measurements, is their sum greater than the third?

No, then no triangle, if yes, then there is a triangle.

2nd Type Problem

Given two sides, what is the range of values possible for the third side.

The range is found by finding the sum and difference between the two given measurements.

Ex: Find the range of the 3rd side if two sides measure 12 and 15.

The range of the 3rd side is between 3 (15 – 12) and 27 (15 + 12).

5.5 Inequalities Involving 2 Triangles

Background

Everything we’ve done up to this point deals with inequalities in one triangle.Triangle Inequality Theorem – Sum of any two sides is greater than the 3rd side.Opposite Angle Theorem – Says that the largest angle is always opposite the largest side.Opposite Side Theorem – Says that the largest side is always opposite the largest angle.

Inequalities in 2 Δ’s

There are two theorems that deals with inequalities in 2 triangles.

SSS Inequality –

SAS Inequality –

SSS Inequality

SSS Inequality (Hinge Theorem) – States that if two sides of one triangle are congruent to two sides of another triangle, then the relationship of the third set of sides governs the relationship of the included angles.

A C

B

D E

F

30 35

Since BC < FE then m<A < m<D.

SAS Inequality

SAS Inequality – States that if two sides of one triangle are congruent to two sides of another triangle, then the relationship of the included angles governs the relationship of the sides opposite those included angles.

A C

B

D E

F

30° 35°

Since m<A < m<D then BC < FE