Proving the Pythagorean Theorem

ID: jabag-jikap
Created by Illustrative MathIllustrative Math
Subject: Geometry
Grade: 9-12

Proving the Pythagorean Theorem

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Problem 1

1) Which of the following is a right triangle?

a) Triangle ABC with AC=6BC=9, and AB=12\text{Triangle } ABC \text{ with } AC = 6 \text{, } BC = 9 \text{, and } AB = 12 b) Triangle DEF with DE=8EF=10, and DF=13\text{Triangle } DEF \text{ with } DE = 8 \text{, } EF = 10 \text{, and } DF = 13 c) Triangle GHI with GI=9HI=12, and GH=15\text{Triangle } GHI \text{ with } GI = 9 \text{, } HI = 12 \text{, and } GH = 15 d) Triangle JKL with JL=10KL=13, and JL=17\text{Triangle } JKL \text{ with } JL = 10 \text{, } KL = 13 \text{, and } JL = 17
Problem 2

In right triangle ABCABC, a square is drawn on each of its sides. An altitude CDCD is drawn to the hypotenuse ABAB and extended to the opposite side of the square on FEFE. In an earlier problem, we discussed Elena’s observation that a2=xca^2 = xc and Diego’s observation that b2=ycb^2 = yc. Mai observes that these statements can be thought of as claims about the areas of rectangles.

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2) Which rectangle has the same area as BGHCBGHC?

3) Which rectangle has the same area as ACIJACIJ?

Problem 3

Andre says he can find the length of the third side of triangle and it is 5 units. Mai disagrees and thinks that the side length is unknown.

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4) Which of them is correct?

a) Andre is correct.\text{Andre is correct.}b) Mai is correct.\text{Mai is correct.}c) They are both correct.\text{They are both correct.}d) They are both incorrect.\text{They are both incorrect.}

5) Show or explain your reasoning.

Problem 4

6) 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) Triangle ACDACD \quad\quad b) Triangle BCDBCD \quad\quad c) Triangle CDBCDB \quad\quad d) Triangle CBDCBD \quad\quad e) Triangle DACDAC

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Problem 5

7) In right triangle ABCABC, altitude CDCD with length 6 is drawn to its hypotenuse. We also know AD=12AD = 12. What is the length of DBDB?

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a) 12 \frac{1}{2} b) 3\text{3}c) 4\text{4}d) 6\text{6}
Problem 6

8) Lines BCBC and DEDE are both vertical. What is the length of BDBD?

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a) 4.5\text{4.5}b) 5\text{5}c) 6\text{6}d) 7.5\text{7.5}
Problem 7

In right triangle ABCABC, AC=5AC = 5 and BC=12BC = 12. A new triangle is formed by connecting the midpoints of ACAC and BCBC.

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9) What is the area of triangle ABCABC?

10) What is the area of triangle DECDEC?

11) Does the scale factor for the side lengths apply to the area as well?

True or false? Write below.

12) If so, explain. If not, what is the scale factor for the area?

Problem 8

13) Quadrilaterals QQ and PP are similar. What is the scale factor of the dilation that takes QQ to PP?

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a) 25 \frac{2}{5} b) 35 \frac{3}{5} c) 45 \frac{4}{5} d) 54 \frac{5}{4}
Problem 9

Priya is trying to determine if triangle ADCADC is congruent to triangle CBACBA. She knows that segments ABAB and DCDC are congruent. She also knows that angles DCADCA and BACBAC are congruent.

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14) Does she have enough information to determine that the triangles are congruent?

True or false? Write below.

15) Explain your reasoning.