# Proving the Pythagorean Theorem

ID: jabag-jikap Illustrative Math
Subject: Geometry

# Proving the Pythagorean Theorem

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##### Problem 1

1) Which of the following is a right triangle?

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

In right triangle ﻿$ABC$﻿, a square is drawn on each of its sides. An altitude ﻿$CD$﻿ is drawn to the hypotenuse ﻿$AB$﻿ and extended to the opposite side of the square on ﻿$FE$﻿. In an earlier problem, we discussed Elena’s observation that ﻿$a^2 = xc$﻿ and Diego’s observation that ﻿$b^2 = yc$﻿. Mai observes that these statements can be thought of as claims about the areas of rectangles. 2) Which rectangle has the same area as ﻿$BGHC$﻿?

3) Which rectangle has the same area as ﻿$ACIJ$﻿?

##### 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. 4) Which of them is correct?

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

5) Show or explain your reasoning.

##### Problem 4

6) 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) Triangle ﻿$ACD$﻿ ﻿$\quad\quad$﻿ b) Triangle ﻿$BCD$﻿ ﻿$\quad\quad$﻿ c) Triangle ﻿$CDB$﻿ ﻿$\quad\quad$﻿ d) Triangle ﻿$CBD$﻿ ﻿$\quad\quad$﻿ e) Triangle ﻿$DAC$﻿ ##### Problem 5

7) In right triangle ﻿$ABC$﻿, altitude ﻿$CD$﻿ with length 6 is drawn to its hypotenuse. We also know ﻿$AD = 12$﻿. What is the length of ﻿$DB$﻿? a) $\frac{1}{2}$b) $\text{3}$c) $\text{4}$d) $\text{6}$
##### Problem 6

8) Lines ﻿$BC$﻿ and ﻿$DE$﻿ are both vertical. What is the length of ﻿$BD$﻿? a) $\text{4.5}$b) $\text{5}$c) $\text{6}$d) $\text{7.5}$
##### Problem 7

In right triangle ﻿$ABC$﻿, ﻿$AC = 5$﻿ and ﻿$BC = 12$﻿. A new triangle is formed by connecting the midpoints of ﻿$AC$﻿ and ﻿$BC$﻿. 9) What is the area of triangle ﻿$ABC$﻿?

10) What is the area of triangle ﻿$DEC$﻿?

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 ﻿$Q$﻿ and ﻿$P$﻿ are similar. What is the scale factor of the dilation that takes ﻿$Q$﻿ to ﻿$P$﻿? a) $\frac{2}{5}$b) $\frac{3}{5}$c) $\frac{4}{5}$d) $\frac{5}{4}$
##### Problem 9

Priya is trying to determine if triangle ﻿$ADC$﻿ is congruent to triangle ﻿$CBA$﻿. She knows that segments ﻿$AB$﻿ and ﻿$DC$﻿ are congruent. She also knows that angles ﻿$DCA$﻿ and ﻿$BAC$﻿ are congruent. 14) Does she have enough information to determine that the triangles are congruent?

True or false? Write below.