A Special Point

ID: kohol-zovaf
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
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15 questions

A Special Point

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Problem 1

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

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a)

α\alpha=β\beta

b)

α>β\alpha>\beta

c)

α\alpha<β\beta

d)

There is not enough information to answer the question.

2) Explain your reasoning.

Problem 2

3) Triangle ABCABC is shown together with its angle bisectors. Draw a point DD that is equidistant from sides ACAC and BCBC, but which is closest to side ABAB.

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Problem 3

4) In triangle ABCABC, point DD is the incenter. Sketch segments to represent the distance from point DD to the sides of the triangle. How must these distances compare?

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a)

The distance from DD to ABAB is greater than the distance from DD to ACAC and the distance from DD to ACAC is greater than

the distance from DD to BCBC.

b)

The distance from DD to ACAC is greater than the distance from DD to BCBC and the distance from DD to BCBC is greater than

the distance from DD to ABAB.

c)

The distance from DD to BCBC is greater than the distance from DD to ABAB and the distance from DD to ABAB is greater than

the distance from DD to ACAC.

d)

The distances must be equal.

e)

There is not enough information to answer the question.

Problem 4

Triangle ABCABC has circumcenter DD.

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

6) If the distance from point DD to point AA is 5 units, what is the distance from point DD to point CC?

7) Explain or show your reasoning.

Problem 5

8) The angles of triangle ABCABC 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

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.

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9) Which student is correct?

a)

Tyler is correct.

b)

Kiran is correct.

c)

Both are correct.

d)

Neither is correct.

10) Explain or show your reasoning.

Problem 7

Given the figure, find the indicated angle measures.

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11) The measure of angle YXWYXW =

12) The measure of angle YZWYZW =

13) The measure of angle XWZXWZ =

Problem 8

14) Which expression describes a point that partitions a segment ABAB in a 1:51:5 ratio?

a)

15A\frac{1}{5}A+45B\frac{4}{5}B

b)

16A\frac{1}{6}A+56B\frac{5}{6}B

c)

45A\frac{4}{5}A+15B\frac{1}{5}B

d)

56A\frac{5}{6}A+16B\frac{1}{6}B

Problem 9

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

a) arcsin(4873)\arcsin \left(\frac{48}{73}\right)\quad\quad b) arccos(4873)\arccos \left(\frac{48}{73}\right) \quad\quad c) arcsin(5573)\arcsin \left(\frac{55}{73}\right) \quad\quad d) arccos(5573)\arccos \left(\frac{55}{73}\right)\quad\quad  e) arctan(5548)\arctan \left(\frac{55}{48}\right)\quad\quad f) arctan(4855)\arctan \left(\frac{48}{55}\right)

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