Lesson 6

A Special Point

  • Let’s see what we can learn about a triangle by watching how salt piles up on it.

Problem 1

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

Quadrilateral A E D F with line A D bisecting angle A into two angles, alpha and beta.

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 \(AB\).

Triangle A B C with angle bisectors that intersect in the center of the triangle.

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?

Triangle ABC with incenter D

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.
Triangle ABC with circumcenter D


(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?


inside the triangle


on the triangle


outside the triangle

(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.

Parallelogram with one angle measuring 20 degrees.

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

(From Unit 7, Lesson 4.)

Problem 7

Find the measures of the remaining angles of quadrilateral \(WXYZ\).

Circle with center W. Radii W X and W Z shown. Segments X Y and Y Z are tangent to the circle. Angle X Y Z is 70 degrees.
(From Unit 7, Lesson 3.)

Problem 8

Which expression describes a point that partitions a segment \(AB\) in a \(1:5\) ratio?


\(\frac15 A+\frac45 B\)


\(\frac16 A+\frac56 B\)


\(\frac45 A+\frac15 B\)


\(\frac56 A+\frac16 B\)

(From Unit 6, Lesson 15.)

Problem 9

Write 3 expressions that can be used to find angle \(C\)

Right triangle abc. Side AB =48 units, side BC=55 units, side CA = 73 units 
(From Unit 4, Lesson 9.)