# Proving the Pythagorean Theorem

# Proving the Pythagorean Theorem

##### Problem 1

1) Which of the following is a right triangle?

##### 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?

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 $DB$?

##### Problem 6

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

##### 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?

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 $P$?

##### 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?

15) Explain your reasoning.