warm-up hair accessories ebony is following directions for folding a handkerchief to make a bandana...

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Warm-up HAIR ACCESSORIES 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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Warm-upHAIR ACCESSORIES 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?

KITE ASSEMBLY Tanya is following directions for making a kite. She has two congruent triangular pieces of fabric that need to be sewn together along their longest side. The directions say to begin sewing the two pieces of fabric together at their smallest angles. At which two angles should she begin sewing?

Answer: A and D

Unit 4 Lesson 4

Special Segments in Triangles and their intersections

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.

a. all angles whose measures are less than m4

b. all angles whose measures are greater than m8Answer: 5, 2, 8, 7

Answer: 4, 9, 5

Perpendicular Bisector

RECALL

Any point on the perpendicular bisector is the same distance from the endpoints of the segment.

Same Distance

RECALL: Theorems

• Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

• Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.

Perpendicular Bisector of a triangle:

• A perpendicular bisector of a side of a triangle is a line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side.

Vocabulary

• When three or more lines intersect at a common point, the lines are called concurrent lines and their point of intersection is call the point of concurrency.

• The point of concurrency of the perpendicular bisectors of triangle is called circumcenter.

Circumcenter Theorem

• The circumcenter of a triangle is equidistant from the vertices of the triangle.

In a triangle there are three perpendicular bisectors. Where they meet is called the…

Circumcenter

The circumcenter is the same distance to each of the vertices.

Example

A. –5

B. 0.5

C. 5

D. 10

In the figure, A is the circumcenter of ΔLMN. Find y if LO = 8y + 9 and ON = 12y – 11.

Example 2

A. 13

B. 11

C. 7

D. –13

In the figure, A is the circumcenter of ΔLMN. Find x if mAPM = 7x + 13.

You Try

A. –12.5

B. 2.5

C. 10.25

D. 12.5

In the figure, A is the circumcenter of ΔLMN. Find r if AN = 4r – 8 and AM = 3(2r – 11).

Vocabulary

• The angle bisectors of a triangle are congruent, and their point of concurrency is called the incenter of a triangle.

Angle BisectorIf you have an angle bisector in a triangle it is the same distance from both the sides of the angle.

A triangle has three angle bisectors.

Where they meet is called the…

Incenter

Given:

Find the measure of angle DGE

Answer:

Use Angle Bisectors

Answer:

Given:

Find the measure of angle ADC

.

Try it!

Example

In the figure, point D is the incenter of ΔABC. What segment is congruent to DG?

___

A. DE

B. DA

C. DC

D. DB

___

___

___

___

Example 2

A. GCD

B. DCG

C. DFB

D. ADE

In the figure, point D is the incenter of ΔABC. What angle is congruent to DCF?

Vocabulary

• A median is a segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.

• The point of concurrency for the medians is called the centriod.

FACT

• The centroid is the point of balance for any triangle.

.

Centroid

Other then the centroid being the center of gravity for a triangle, there is something else special about the centroid.

The centroid is always 2/3 the total distance from the

vertex to the other side.

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

Centroid Theorem

YV = 12

Simplify.

Answer: YP = 8; PV = 4

YP + PV = YV Segment Addition8 + PV = 12 YP = 8

PV = 4 Subtract 8 from each side.

Example 2

If G is the centriod, and GE = 6, how long is BE?

A

G

F

ED

C

B

You try

A. LR = 15; RO = 15

B. LR = 20; RO = 10

C. LR = 17; RO = 13

D. LR = 18; RO = 12

In ΔLNP, R is the centroid and LO = 30. Find LR and RO.

New Vocabulary• The altitude is a segment from the vertex of

triangle to the opposite side that is perpendicular to that side. The intersection point of the altitudes of a triangle is called the

orthocenter.

The everyday meaning of altitude…

The vertical elevation of an object above a surface

ALGEBRA Points U, V, and W are the midpoints of respectively. Find a, b, and c.

Segment Measures

ALGEBRA Points T, H, and G are the midpoints of respectively. Find w, x, and y.

Answer:

Try it!

• Unfortunately, there is nothing special about the orthocenter, but you will still be required to find it.

Vocabulary Tree Map

Point of Concurrency

Perpendicular Bisectors

Circumcenter

Angle Bisectors

Incenter

Medians

Centroid

Altitudes

Orthocenter

Classwork

Classwork