Using the Pythagorean Theorem and Similarity

ID: goboj-jorih
Created by Illustrative MathIllustrative Math
Subject: Geometry
Grade: 9-12

Using the Pythagorean Theorem and Similarity

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Problem 1

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

a) ABCABC \quad\quad b) ACDACD \quad\quad c) BCDBCD \quad\quad d) BDCBDC \quad\quad e) CADCAD \quad\quad f) CBDCBD

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Problem 2

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

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Problem 3

3) In triangle ABCABC (not a right triangle), altitude CDCD is drawn to side ABAB. The length of ABAB is cc. Which of the following statements must be true?

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a) The measure of angle ACB is the same measure as angle B.\text{The measure of angle } ACB \text{ is the same measure as angle } B \text{.}b) b2 = c2 + a2 b^2 \ = \ c^2 \ +\ a^2 c) Triangle ADC is similar to triangle ACB.\text{Triangle } ADC \text{ is similar to triangle } ACB \text{.}d) The area of triangle ABC equals 12hc\text{The area of triangle } ABC \text{ equals } \frac{1}{2}h\cdot c
Problem 4

4) Quadrilateral ABCDABCD is similar to quadrilateral ABCDA'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) ABAB=ACAC\frac{A'B'}{AB} = \frac{A'C'}{AC} \quad\quad b) ABAB=ACAC\frac{A'B'}{AB} = \frac{AC}{A'C'} \quad\quad c) ABCD=ABCD\frac{A'B'}{C'D'} = \frac{AB}{CD} \quad\quad d) ADAD=BCBC\frac{AD}{A'D'} = \frac{BC}{B'C'} \quad\quad e) ABAD=ADAB\frac{AB}{A'D'} = \frac{AD}{A'B'}

Problem 5

Segment ABA'B' is parallel to segment ABAB.

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

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

Problem 6

7) Lines BCBC and DEDE are both vertical. What is the length of ADAD?

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

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

a) Triangle BDEBDE is congruent to triangle FCEFCE \quad\quad b) Triangle BDEBDE is congruent to triangle FDAFDA

c) BDBD is congruent to FEFE \quad\quad d) The length of BCBC is 8 \quad\quad e) The length of BCBC is 6

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