Lesson 14

Proving the Pythagorean Theorem

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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

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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 \(ACIJ\)?
Right triangle A B C with squares drawn on each side. 

Solution

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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. 

Triangle A B C. Length of A B is 4 and A C is 3.

Solution

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Problem 4

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

Right triangle A B C has right angle C and altitude C D drawn to the hypotenuse. Ange B D C is a right angle.

Solution

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(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 \(DB\)?

Right triangle A B C has right angle C and altitude C D drawn to the hypotenuse A B. Angle B D C is a right angle. C D is 6 and A D is 12.
A:

\(\frac12\)

B:

3

C:

4

D:

6

Solution

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(From Unit 3, Lesson 13.)

Problem 6

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

Line A B is horizontal and has 3 points, A, B and D. Lines B C and D E are vertical. Line A C goes through C on B C and E on D E. A B is 3, B C is 2 and D E is 5.  
A:

4.5

B:

5

C:

6

D:

7.5

Solution

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(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 \(BC\).

Right triangles A B C and D E C. D midpoint of A C. E midpoint of B C. A C labeled 5. B C labeled 12.
  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

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(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 \(P\)?

Two concave quadrilaterals labeled P and Q.
A:

\(\frac25\)

B:

\(\frac35\)

C:

\(\frac45\)

D:

\(\frac54\)

Solution

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(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.

Triangle ABC and ACD.

Solution

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(From Unit 2, Lesson 6.)