# Lesson 6

A Special Point

### Problem 1

How do the values of $$\alpha$$ and $$\beta$$ compare? Explain your reasoning.

### Problem 2

Triangle $$ABC$$ is shown together with its angle bisectors. Draw a point $$D$$ that is equidistant from sides $$AC$$ and $$BC$$, but which is closest to side $$AB$$.

### Problem 3

In triangle $$ABC$$, point $$D$$ is the incenter. Sketch segments to represent the distance from point $$D$$ to the sides of the triangle. How must these distances compare?

### Problem 4

Triangle $$ABC$$ has circumcenter $$D$$.

1. Sketch the 3 lines that intersect at the circumcenter.
2. If the distance from point $$D$$ to point $$A$$ is 5 units, what is the distance from point $$D$$ to point $$C$$? Explain or show your reasoning.

### Solution

(From Unit 7, Lesson 5.)

### Problem 5

The angles of triangle $$ABC$$ measure 50 degrees, 40 degrees, and 90 degrees. Will its circumcenter fall inside the triangle, on the triangle, or outside the triangle?

A:

inside the triangle

B:

on the triangle

C:

outside the triangle

### Solution

(From Unit 7, Lesson 5.)

### Problem 6

Tyler and Kiran are discussing the parallelogram in the image. Tyler says the parallelogram cannot be cyclic. Kiran says the parallelogram can be cyclic if a circle is drawn carefully through the vertices.

Do you agree with either of them? Explain or show your reasoning.

### Solution

(From Unit 7, Lesson 4.)

### Problem 7

Find the measures of the remaining angles of quadrilateral $$WXYZ$$.

### Solution

(From Unit 7, Lesson 3.)

### Problem 8

Which expression describes a point that partitions a segment $$AB$$ in a $$1:5$$ ratio?

A:

$$\frac15 A+\frac45 B$$

B:

$$\frac16 A+\frac56 B$$

C:

$$\frac45 A+\frac15 B$$

D:

$$\frac56 A+\frac16 B$$

### Solution

Write 3 expressions that can be used to find angle $$C$$