# Circles in Triangles

Illustrative Mathematics, CC BY 4.0
Subject: Geometry

13 questions

# Circles in Triangles

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##### Problem 1

Triangle ﻿$ABC$﻿ is shown with its incenter at ﻿$D$﻿. The inscribed circle’s radius measures 2 units. The length of ﻿$AB$﻿ is 9 units. The length of ﻿$BC$﻿ is 10 units. The length of ﻿$AC$﻿ is 17 units.

1) What is the area of triangle ﻿$ACD$﻿?

2) What is the area of triangle ﻿$ABC$﻿?

##### Problem 2

3) Triangle ﻿$ABC$﻿ is shown with an inscribed circle of radius 4 units centered at point ﻿$D$﻿. The inscribed circle is tangent to side ﻿$AB$﻿ at the point ﻿$G$﻿. The length of ﻿$AG$﻿ is 6 units and the length of ﻿$BG$﻿ is 8 units. What is the measure of angle ﻿$A$﻿?

a)

﻿$\arctan\left(\frac{2}{3}\right)$﻿

b)

2﻿$\arctan\left(\frac{2}{3}\right)$﻿

c)

﻿$\arcsin\left(\frac{2}{3}\right)$﻿

d)

2﻿$\arccos\left(\frac{2}{3}\right)$﻿

##### Problem 3

4) Construct the inscribed circle for the triangle.

##### Problem 4

Point ﻿$D$﻿ lies on the angle bisector of angle ﻿$ACB$﻿. Point ﻿$E$﻿ lies on the perpendicular bisector of side ﻿$AB$﻿.

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

a) Point ﻿$D$﻿ is the same distance to sides ﻿$AC$﻿ and ﻿$AB$﻿.

b) Point ﻿$D$﻿ is the same distance to sides ﻿$AC$﻿ and ﻿$BC$﻿.

c) Point ﻿$D$﻿ is the same distance to sides ﻿$BC$﻿ and ﻿$AB$﻿.

d) Point ﻿$D$﻿ is equidistant from ﻿$A$﻿ and ﻿$B$﻿.

e) Point ﻿$D$﻿ is closer to ﻿$A$﻿ than it is to ﻿$B$﻿.

f) Point ﻿$D$﻿ is closer to ﻿$B$﻿ than it is to ﻿$A$﻿.

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

a) Point ﻿$E$﻿ is equidistant from points ﻿$A$﻿ and ﻿$B$﻿.

b) Point ﻿$E$﻿ is equidistant from points ﻿$C$﻿ and ﻿$B$﻿.

c) Point ﻿$E$﻿ is equidistant from points ﻿$A$﻿ and ﻿$C$﻿.

d) Point ﻿$E$﻿ is closer to side ﻿$AB$﻿ than it is to side ﻿$AC$﻿.

e) Point ﻿$E$﻿ is closer to side ﻿$AC$﻿ than it is to side ﻿$AB$﻿.

f) Point ﻿$E$﻿ is closer to side ﻿$BC$﻿ than it is to side ﻿$AC$﻿

##### Problem 5

7) Construct the incenter of the triangle.

##### Problem 6

The angles of triangle ﻿$ABC$﻿ 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

##### 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.

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