# Lesson 7

Circles in Triangles

- Let’s construct the largest possible circle inside of a triangle.

### 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.

- What is the area of triangle \(ACD\)?
- What is the area of triangle \(ABC\)?

### 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 \(A\)?

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

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

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

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

### Problem 3

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 \(AB\).

- What can we say about the distance between point \(D\) and the sides and vertices of triangle \(ABC\)?
- What can we say about the distance between point \(E\) and the sides and vertices of triangle \(ABC\)?

### Problem 5

Construct the incenter of the triangle. Explain your reasoning.

### 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.

### 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.

### 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?