Lesson 15

Finding All the Unknown Values in Triangles

Problem 1

In the right triangles shown, the measure of angle \(ABC\) is the same as the measure of angle \(EBD\).  ​​​What is the length of side \(BE\)?

Two right triangles, A B C and E B D. Angles C and D are right angles and D is above C on B C. Side A B is 5, A C is 3 and D E is 2. Angle A B C is congruent to angle E B D.

Solution

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Problem 2

In right triangle \(ABC\), angle \(C\) is a right angle, \(AB=13\), and \(BC=5\). What is the length of \(AC\)?

Solution

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Problem 3

In this diagram, lines \(AC\) and \(DE\) are parallel, and line \(DC\) is perpendicular to each of them. What is a reasonable estimate for the length of side \(BE\)?

\(AC\parallel DE\), \(DC \perp DE \), \(DC \perp AC\), segment $DE$ has length 1

Parallel line segments A C and D E. Segment D C is perpendicular to A C and D E. Point B on segment D C, segment A E drawn passing through B. A B labeled 5, B C labeled 4.
A:

\(\frac{1}{3}\)

B:

1

C:

\(\frac{5}{3}\)

D:

5

Solution

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Problem 4

Select all of the right triangles. 

A:

Triangle \(ABC\) with \(AB = 30\), \(BC = 40\), and \(AC = 50\)

B:

Triangle \(XYZ\) with \(XY = 1\), \(YZ = 1\), and \(XZ = 2\)

C:

Triangle \(EFG\) with \(EF = 8\), \(FG = 15\), and \(EG = 17\)

D:

Triangle \(LMN\) with \(LM = 7\), \(MN = 24\), and \(LN = 25\)

E:

Triangle \(QRS\) with \(QR = 4\), \(RS = 5\), and \(QS = 6\)

Solution

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(From Unit 3, Lesson 14.)

Problem 5

Andre says he can find the length of the third side of triangle \(ABC\) and it is 13 units. Mai disagrees and thinks that the side length is unknown. Who do you agree with? Show or explain your reasoning. 

Triangle A B C. Side A B labeled 5. Side A C labeled 12. Diagonal B C, unlabeled. No angles labeled.

Solution

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(From Unit 3, Lesson 14.)

Problem 6

In right triangle \(ABC\), altitude \(CD\) with length \(h\) is drawn to its hypotenuse. We also know \(AD=8\) and \(DB=2\). What is the value of \(h\)?

Right triangle A B C, an altitude C D with length h is drawn to its hypotenuse. 

Solution

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(From Unit 3, Lesson 13.)

Problem 7

Select the sequence of transformations that would show that triangles \(ABC\) and \(AED\) are similar. The length of \(AC\) is 6.

\(AC=6\)

Triangle A B C and A D E. Point D is located on side A C and point E is to the right of side A C. Side A B is 8, B C is 4, A C is 6, A D is 3, A E is 4, and D E is 2.
A:

Dilate from center \(A\) by a scale factor of \(2\), then reflect over line \(AC\).

B:

Dilate from center \(A\) by a scale factor of \(2\), then rotate 60º around angle \(A\).

C:

Translate by directed line segment \(DC\), then reflect over line \(AC\).

D:

Dilate from center \(A\) by a scale factor of \(4\), then reflect over line \(AC\).

Solution

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(From Unit 3, Lesson 7.)