# A Special Point

15 questions

# A Special Point

##### Problem 1

1) How do the values of $\alpha$ and $\beta$ compare?

$\alpha$=$\beta$

$\alpha>\beta$

$\alpha$<$\beta$

There is not enough information to answer the question.

2) Explain your reasoning.

##### Problem 2

3) 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

4) 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?

The distance from $D$ to $AB$ is greater than the distance from $D$ to $AC$ and the distance from $D$ to $AC$ is greater than

the distance from $D$ to $BC$.

The distance from $D$ to $AC$ is greater than the distance from $D$ to $BC$ and the distance from $D$ to $BC$ is greater than

the distance from $D$ to $AB$.

The distance from $D$ to $BC$ is greater than the distance from $D$ to $AB$ and the distance from $D$ to $AB$ is greater than

the distance from $D$ to $AC$.

The distances must be equal.

There is not enough information to answer the question.

##### Problem 4

Triangle $ABC$ has circumcenter $D$.

5) Sketch the 3 lines that intersect at the circumcenter.

6) If the distance from point $D$ to point $A$ is 5 units, what is the distance from point $D$ to point $C$?

7) Explain or show your reasoning.

##### Problem 5

8) 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

##### 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.

9) Which student is correct?

Tyler is correct.

Kiran is correct.

Both are correct.

Neither is correct.

10) Explain or show your reasoning.

##### Problem 7

Given the figure, find the indicated angle measures.

11) The measure of angle $YXW$ =

12) The measure of angle $YZW$ =

13) The measure of angle $XWZ$ =

##### Problem 8

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

$\frac{1}{5}A$+$\frac{4}{5}B$

$\frac{1}{6}A$+$\frac{5}{6}B$

$\frac{4}{5}A$+$\frac{1}{5}B$

$\frac{5}{6}A$+$\frac{1}{6}B$

##### Problem 9

15) Select $\textbf{all}$ expressions that could be used to find angle $C$. Write each corresponding letter in the answer box and separate letters with commas.

a) $\arcsin \left(\frac{48}{73}\right)\quad\quad$ b) $\arccos \left(\frac{48}{73}\right) \quad\quad$ c) $\arcsin \left(\frac{55}{73}\right) \quad\quad$ d) $\arccos \left(\frac{55}{73}\right)\quad\quad$ e) $\arctan \left(\frac{55}{48}\right)\quad\quad$ f) $\arctan \left(\frac{48}{55}\right)$