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5-2 Mathematical Literacy and VocabularyBisectors in Triangles

For Exercises 1–6, match the term in Column A with its description in Column B. The first one is done for you.

Column A Column B

1. concurrent the point of intersection of three or more lines

2. point of concurrency the intersection point of the three angle bisectors of a triangle

3. circumcenter of a triangle when a circle touches the three sides of a triangle

4. circumscribed term that describes three or more lines that intersect at a single point

5. incenter of a triangle when a circle passes through the three vertices of a triangle

6. inscribed the intersection point of the three perpendicular bisectors of a triangle

For Exercises 7–9, match the phrase in Column A with the diagram in Column B that best describes point P.

Column A Column B

7. circumcenter of a triangle P

8. point of concurrency

P

9. incenter of a triangle P

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5-2 ReteachingBisectors in Triangles

1. Write the letter of each figure beside its definition.

The circumcenter is the point of concurrency of the perpendicular bisectors of a triangle.

The circumscribed circle is centered at the circumcenter and contains the vertices of a triangle.

The incenter is the point of concurrency of the angle bisectors of a triangle.

The inscribed circle is centered at the incenter, and the sides of the triangle touch the circle.

2. Timothy says PX = PY = PZ .

a. What mistake did Timothy likely make?

b. From what points is P equidistant? Explain your reasoning.

3. Complete the sentences below to find the value of x.

‾ AP , ‾ BP , and ‾ CP are the of △ABC.

So P is the of △ABC.

The incenter is equidistant from the of △ABC, so PS = .

Therefore, x = .

A. B. C. D.

A

YX

CB Z

P

A

S

B

C

PX

TV

910

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5-2 Additional PracticeBisectors in Triangles

Exercises 1–3. The perpendicular bisectors of △DEF are ‾ TM , ‾ UM , and ‾ VM .

1. What is the circumcenter of △DEF? Explain your reasoning.

2. What do you know about MD, ME, and MF? Explain your reasoning.

3. Name three isosceles triangles.

Exercises 4–5. Points X, Y, and Z are the midpoints of the sides of △ABC.

4. What point is the center of the circle that contains A, B, and C? Explain your reasoning.

5. What point is the center of the circle that intersects each side of △ABC at exactly one point? Explain your reasoning.

6. Understand What is the diameter of the inscribed circle of the triangle?

7. Apply A farmer wants to place a hay bale so that it is the same distance from Gates L, M, and N. Construct the location P for the bale of hay.

D

U

FV

M

T

E

A

ZE

CB F

M

Y

G

X

x + 22x − 3

N

ML

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A

B C

OIJ

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5-2 EnrichmentBisectors in Triangles

In the construction, point O is the circumcenter and point I is the incenter of triangle ABC.

1. The circle with center J is an excircle. Is point J collinear with points I and B? Explain how you know.

2. a. How would you construct the excircle J?

Step 1:

Step 2: Use the intersection point as the of the excircle and construct a tangent to either side of angle or segment AC.

b. The intersection of the angle bisectors of the interior angles of a triangle is called the . Constructing an excircle is like constructing

. Change one word in the definition above to write the definition of an excircle.

3. Write a proof.

Given: △ ABC with angle bisectors ↔

BJ , ↔

AJ , and ↔

C J

Prove: lines ↔

BJ , ↔

AJ , and ↔

C J are concurrent at a point J equidistant from ↔

BC , ↔

BA , and ‾ AC .

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