# Using the Pythagorean Theorem and Similarity

ID: goboj-jorih
Illustrative Math
Subject: Geometry

# Using the Pythagorean Theorem and Similarity

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##### Problem 1

1) In right triangle ﻿$ABC$﻿, altitude ﻿$CD$﻿ is drawn to its hypotenuse. Select all triangles which must be similar to triangle ﻿$ABC$﻿. Write each corresponding letter in the answer box and separate letters with commas.

a) ﻿$ABC$﻿ ﻿$\quad\quad$﻿ b) ﻿$ACD$﻿ ﻿$\quad\quad$﻿ c) ﻿$BCD$﻿ ﻿$\quad\quad$﻿ d) ﻿$BDC$﻿ ﻿$\quad\quad$﻿ e) ﻿$CAD$﻿ ﻿$\quad\quad$﻿ f) ﻿$CBD$﻿

##### Problem 2

2) In right triangle ﻿$ABC$﻿, altitude ﻿$CD$﻿ with length ﻿$h$﻿ is drawn to its hypotenuse. We also know ﻿$AD = 12$﻿ and ﻿$DB = 3$﻿. What is the value of ﻿$h$﻿?

##### Problem 3

3) In triangle ﻿$ABC$﻿ (not a right triangle), altitude ﻿$CD$﻿ is drawn to side ﻿$AB$﻿. The length of ﻿$AB$﻿ is ﻿$c$﻿. Which of the following statements must be true?

a) $\text{The measure of angle } ACB \text{ is the same measure as angle } B \text{.}$b) $b^2 \ = \ c^2 \ +\ a^2$c) $\text{Triangle } ADC \text{ is similar to triangle } ACB \text{.}$d) $\text{The area of triangle } ABC \text{ equals } \frac{1}{2}h\cdot c$
##### Problem 4

4) Quadrilateral ﻿$ABCD$﻿ is similar to quadrilateral ﻿$A'B'C'D'$﻿. Select all equations that could be used to solve for missing lengths. Write each corresponding letter in the answer box and separate letters with commas.

a) ﻿$\frac{A'B'}{AB} = \frac{A'C'}{AC}$﻿ ﻿$\quad\quad$﻿ b) ﻿$\frac{A'B'}{AB} = \frac{AC}{A'C'}$﻿ ﻿$\quad\quad$﻿ c) ﻿$\frac{A'B'}{C'D'} = \frac{AB}{CD}$﻿ ﻿$\quad\quad$﻿ d) ﻿$\frac{AD}{A'D'} = \frac{BC}{B'C'}$﻿ ﻿$\quad\quad$﻿ e) ﻿$\frac{AB}{A'D'} = \frac{AD}{A'B'}$﻿

##### Problem 5

Segment ﻿$A'B'$﻿ is parallel to segment ﻿$AB$﻿.

5) What is the length of segment ﻿$A'A$﻿?

6) What is the length of segment ﻿$B'B$﻿?

##### Problem 6

7) Lines ﻿$BC$﻿ and ﻿$DE$﻿ are both vertical. What is the length of ﻿$AD$﻿?

a) $\text{4.5}$b) $\text{5}$c) $\text{7.5}$d) $\text{10}$
##### Problem 7

8) Triangle ﻿$DEF$﻿ is formed by connecting the midpoints of the sides of triangle ﻿$ABC$﻿. Select all true statements. Write each corresponding letter in the answer box and separate letters with commas.

a) Triangle ﻿$BDE$﻿ is congruent to triangle ﻿$FCE$﻿ ﻿$\quad\quad$﻿ b) Triangle ﻿$BDE$﻿ is congruent to triangle ﻿$FDA$﻿

c) ﻿$BD$﻿ is congruent to ﻿$FE$﻿ ﻿$\quad\quad$﻿ d) The length of ﻿$BC$﻿ is 8 ﻿$\quad\quad$﻿ e) The length of ﻿$BC$﻿ is 6