Lesson 7

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.

Triangle ABC with a circle inside with center D.
  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 \(A\)?

Triangle ABC with a circle inside with center D. Dashed lines from D to outside of circle, one labeled 4. AG = 6. BG = 8.
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.

Triangle ABC

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 \(ABC\)?
Triangle A B C. Dotted angle bisector drawn from C to A B, with point D. Second dotted angle bisector, drawn to right of other bisector, passing through A B and B C, with point E.

 

Solution

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(From Unit 7, Lesson 6.)

Problem 5

Construct the incenter of the triangle. Explain your reasoning.

Triangle ABC

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.

park A

Quadrilateral with interior angles =57, 128, 100, and 75 degrees 

park B

Quadrilateral with interior angles = 75, 133, 105, and 47 degrees 

 

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.)