# Lesson 7

Circles in Triangles

### 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 $$ABC$$?

### Solution

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### Problem 2

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:

$$\arctan\left(\frac23\right)$$

B:

$$2\arctan\left(\frac23\right)$$

C:

$$\arcsin\left(\frac23\right)$$

D:

$$2 \arccos\left(\frac23\right)$$

### Solution

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### Problem 3

Construct the inscribed circle for the triangle.

### Solution

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### Problem 4

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

1. What can we say about the distance between point $$D$$ and the sides and vertices of triangle $$ABC$$?
2. What can we say about the distance between point $$E$$ and the sides and vertices of triangle $$ABC$$?

### Solution

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(From Unit 7, Lesson 6.)

### Problem 5

Construct the incenter of the triangle. Explain your reasoning.

### Solution

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(From Unit 7, Lesson 6.)

### Problem 6

The angles of triangle $$ABC$$ measure 30 degrees, 40 degrees, and 110 degrees. Will its circumcenter fall inside the triangle, on the triangle, or outside the triangle? Explain your reasoning.

### Solution

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(From Unit 7, Lesson 5.)

### 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. Is this possible for either park? Explain or show your reasoning.

### Solution

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(From Unit 7, Lesson 4.)

### Problem 8

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

### Solution

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(From Unit 6, Lesson 16.)