Name: Chapter 5: Relationships in Triangles Lesson 5-1...

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Name: Chapter 5: Relationships in Triangles Page 1 Lesson 5-1: Bisectors of Triangles Date: A is a line, segment, or ray that is perpendicular to the given segment and passes through its midpoint. Example: Example 1: Use the Perpendicular Bisector Theorems A. Find B. Find C. Find

Transcript of Name: Chapter 5: Relationships in Triangles Lesson 5-1...

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Lesson 5-1: Bisectors of Triangles Date:

A is a line, segment, or ray that is perpendicular

to the given segment and passes through its midpoint.

Example:

Example 1: Use the Perpendicular Bisector Theorems

A. Find 𝐵𝐶 B. Find 𝑋𝑌 C. Find 𝑃𝑄

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When three or more lines intersect , they are called .

The point where these lines intersect is called the .

Since a triangle has three sides, it can have perpendicular bisectors.

There point where the three perpendicular bisectors intersect (their point of concurrency) is called the

of the triangle.

Example 2: Use the Circumcenter Theorem

GARDEN A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be

inside the garden?

There can be three locations for the circumcenter:

Interior Exterior Side

Acute Triangle Obtuse Triangle Right Triangle

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Example 2B: Point 𝑃 is the circumcenter of ∆𝐸𝑀𝐾. List any segments congruent to the segments below.

a. 𝑀𝑌̅̅̅̅̅

b. 𝐾𝑃̅̅ ̅̅

c. 𝑀𝑁̅̅ ̅̅ ̅

d. 𝐸𝑅̅̅ ̅̅

An divides an angle into two congruent angles.

Example 3: Use the Angle Bisector Theorems

A. Find 𝐷𝐵 B. Find 𝑚∠𝑊𝑌𝑍 C. Find 𝑄𝑆

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Since a triangle has three angles, it can have angle bisectors.

The point where the three angle bisectors intersect (their point of concurrency) is called the

of a triangle.

Example 4: Use the Incenter Theorem

A. Find 𝑆𝑇 if 𝑆 is the incenter of ∆𝑀𝑁𝑃. B. Find 𝑚∠𝑆𝑃𝑈 if 𝑆 is the incenter of ∆𝑀𝑁𝑃.

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Lesson 5-2: Medians and Altitudes of Triangles Date:

A of a triangle is a segment with endpoints being a vertex of a

triangle and the midpoint of the opposite side.

Since a triangle has three sides and angles, every triangle has three medians that are concurrent.

The point of concurrency of the medians of a triangle is called the and is

always inside the triangle.

Example 1: In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.

Example 2: In ΔABC, CG = 4. Find GE.

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Example 3: SCULPTURE An artist is designing a sculpture that balances a triangle on top of a pole. In the artist’s

design on the coordinate plane, the vertices are located at (1, 4), (3, 0), and (3, 8). What are the coordinates of the

point where the artist should place the pole under the triangle so that it will balance?

An of a triangle is a segment from a vertex to the line containing

the opposite side and perpendicular to the line containing that side.

An altitude can lie in the interior, exterior, or on the side of a triangle.

Since a triangle has three sides and angles, every triangle has three altitudes that are concurrent.

The point of concurrency of the altitudes of a triangle is called the .

Example 4: COORDINATE GEOMETRY The vertices of ΔHIJ are H(1, 2), I(–3, –3), and J(–5, 1). Find the

coordinates of the orthocenter of ΔHIJ.

x

y

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Lesson 5-3: Inequalities in One Triangle Date:

Example 1:

A. Use the Exterior Angle Inequality

Theorem to list all angles whose measures

are less than 𝑚∠14.

B. Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than 𝑚∠5.

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Example 2: List the angles of ΔABC in order from smallest to largest.

Example 3: List the sides of ΔABC in order from shortest to longest.

Example 4: Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds

the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds

the handkerchief with the dimensions shown, which two ends should she tie?

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Lesson 5-5: The Triangle Inequality Date:

Example 1: Is it possible to form a triangle with the given side lengths? If not, explain.

A. 61

2, 6

1

2, and 14

1

2 B. 6.8, 7.2, 5.1

Example 2: In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR?

a. 7

b. 9

c. 11

d. 13

Example 3: The towns of Jefferson, Kingston, and Newbury are shown in the map below. Prove that

driving first from Jefferson to Kingston and then Kingston to Newbury is a greater distance than driving

from Jefferson to Newbury.

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Lesson 5-6: Inequalities in Two Triangles Date:

Example 1:

A. Compare the measures AD and BD. B. Compare the measures mABD and mBDC.

Example 2: Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back.

The patient lies flat on the examining table, and the doctor raises each leg until the patient experiences pain

in the back area. Nick can tolerate the doctor raising his right leg 35° and his left leg 65° from the table.

Which leg can Nick raise higher above the table?

Example 3: Find the range of possible values for a.