# Lesson 14

Proving the Pythagorean Theorem

### Problem 1

Which of the following are right triangles?

A:

Triangle $$ABC$$ with $$AC=6$$, $$BC=9$$, and $$AB=12$$

B:

Triangle $$DEF$$ with $$DE=8$$, $$EF=10$$, and $$FD=13$$

C:

Triangle $$GHI$$ with $$GI=9$$, $$HI=12$$, and $$GH=15$$

D:

Triangle $$JKL$$ with $$JL=10$$, $$KL=13$$, and $$JL=17$$

### Solution

For access, consult one of our IM Certified Partners.

### 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 class, 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.

1. Which rectangle has the same area as $$BGHC$$?
2. Which rectangle has the same area as $$ACIJ$$?

### Solution

For access, consult one of our IM Certified Partners.

### Problem 3

Andre says he can find the length of the third side of triangle $$ABC$$ and it is 5 units. Mai disagrees and thinks that the side length is unknown. Do you agree with either of them? Show or explain your reasoning.

### Solution

For access, consult one of our IM Certified Partners.

### Problem 4

In right triangle $$ABC$$, altitude $$CD$$ is drawn to its hypotenuse. Find 2 triangles which must be similar to triangle $$ABC$$.

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 3, Lesson 13.)

### Problem 5

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:

$$\frac12$$

B:

3

C:

4

D:

6

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 3, Lesson 13.)

### Problem 6

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

A:

4.5

B:

5

C:

6

D:

7.5

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 3, Lesson 12.)

### Problem 7

In right triangle $$ABC$$, $$AC=5$$ and $$BC=12$$. A new triangle $$DEC$$ is formed by connecting the midpoints of $$AC$$ and $$BC$$.

1. What is the area of triangle $$ABC$$?
2. What is the area of triangle $$DEC$$?
3. Does the scale factor for the side lengths apply to the area as well?

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 3, Lesson 11.)

### Problem 8

Quadrilaterals $$Q$$ and $$P$$ are similar.

What is the scale factor of the dilation that takes $$Q$$ to $$P$$?

A:

$$\frac25$$

B:

$$\frac35$$

C:

$$\frac45$$

D:

$$\frac54$$

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 3, Lesson 6.)

### 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. Does she have enough information to determine that the triangles are congruent? Explain your reasoning.

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 2, Lesson 6.)