Circles in Triangles

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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
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13 questions

Circles in Triangles

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Problem 1

Triangle ABCABC is shown with its incenter at DD. The inscribed circle’s radius measures 2 units. The length of ABAB is 9 units. The length of BCBC is 10 units. The length of ACAC is 17 units.

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1) What is the area of triangle ACDACD?

2) What is the area of triangle ABCABC?

Problem 2

3) Triangle ABCABC is shown with an inscribed circle of radius 4 units centered at point DD. The inscribed circle is tangent to side ABAB at the point GG. The length of AGAG is 6 units and the length of BGBG is 8 units. What is the measure of angle AA?

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a)

arctan(23)\arctan\left(\frac{2}{3}\right)

b)

2arctan(23)\arctan\left(\frac{2}{3}\right)

c)

arcsin(23)\arcsin\left(\frac{2}{3}\right)

d)

2arccos(23)\arccos\left(\frac{2}{3}\right)

Problem 3

4) Construct the inscribed circle for the triangle.

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Problem 4

Point DD lies on the angle bisector of angle ACBACB. Point EE lies on the perpendicular bisector of side ABAB.

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5) Select all\textbf{all} statements that must\textbf{must} be true about the distance between point DD and the sides and vertices of triangle ABCABC. Write each corresponding letter in the answer box and separate letters with commas.

a) Point DD is the same distance to sides ACAC and ABAB.

b) Point DD is the same distance to sides ACAC and BCBC.

c) Point DD is the same distance to sides BCBC and ABAB.

d) Point DD is equidistant from AA and BB.

e) Point DD is closer to AA than it is to BB.

f) Point DD is closer to BB than it is to AA.

6) Select all\textbf{all} statements that must\textbf{must} be true about the distance between point EE and the sides and vertices of triangle ABCABC. Write each corresponding letter in the answer box and separate letters with commas.

a) Point EE is equidistant from points AA and BB.

b) Point EE is equidistant from points CC and BB.

c) Point EE is equidistant from points AA and CC.

d) Point EE is closer to side ABAB than it is to side ACAC.

e) Point EE is closer to side ACAC than it is to side ABAB.

f) Point EE is closer to side BCBC than it is to side ACAC

Problem 5

7) Construct the incenter of the triangle.

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8) Explain your reasoning.

Problem 6

The angles of triangle ABCABC measure 30 degrees, 40 degrees, and 110 degrees.

9) Where will its circumcenter fall?

a)

inside the triangle

b)

outside the triangle

c)

on the triangle

d)

impossible to tell from the information given

10) Explain your reasoning.

Problem 7

The images show 2 possible blueprints for a park. The park planners want to build a water fountain that is equidistant from each of the corners of the park.

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11) Is this possible for either park?

a)

It is possible for Park A, but not for Park B.

b)

It is possible for Park B, but not for Park A.

c)

It is possible for both parks.

d)

It is not possible for either park.

12) Explain or show your reasoning.

Problem 8

13) Triangle ABCABC has vertices at (8,2)(-8,2), (2,6)(2,6), and (10,2)(10,2). What is the point of intersection of the triangle’s medians?