Lesson 13

Using the Pythagorean Theorem and Similarity

  • Let’s explore right triangles with altitudes drawn to the hypotenuse.

Problem 1

In right triangle \(ABC\), altitude \(CD\) is drawn to its hypotenuse. Select all 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.
A:

\(ABC\)

B:

\(ACD\)

C:

\(BCD\)

D:

\(BDC\)

E:

\(CAD\)

F:

\(CBD\)

Problem 2

In right triangle \(ABC\), altitude \(CD\) with length \(h\) is drawn to its hypotenuse. We also know \(AD=12\) and \(DB=3\). What is the value of \(h\)?

Right triangle A B C, angle C marked as right. Altitude C D drawn to hypotenuse A B. Segment C D labeled h. Segment B D labeled 3. Segment D A labeled 12.

Problem 3

In triangle \(ABC\) (not a right triangle), altitude \(CD\) is drawn to side \(AB\). The length of \(AB\) is \(c\). Which of the following statements must be true?

Triangle A B C is not a right triangle. Altitude C D is drawn to side A B and is length h. Side A C is b, side B C is a and A D is e. Angle A C B is 75 degrees. Side A B is length c.
A:

The measure of angle \(ACB\) is the same measure as angle \(B\).

B:

\(b^2=c^2+a^2\).

C:

Triangle \(ADC\) is similar to triangle \(ACB\).

D:

The area of triangle \(ABC\) equals \(\frac{1}{2}h\boldcdot c\).

Problem 4

Quadrilateral \(ABCD\) is similar to quadrilateral \(A’B’C’D’\). Write 2 equations that could be used to solve for missing lengths. 

(From Unit 3, Lesson 12.)

Problem 5

Segment \(A’B’\) is parallel to segment \(AB\).

  1. What is the length of segment \(A'A\)?
  2. What is the length of segment \(B’B\)?
Triangles A B C and A prime B prime C. Lengths as follows: Directed segment A B, 20. Directed segment A prime B prime, 8. Segment B prime C, 6. Segment A prime C, 12.
(From Unit 3, Lesson 11.)

Problem 6

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

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:

7.5

D:

10

(From Unit 3, Lesson 12.)

Problem 7

Triangle \(DEF\) is formed by connecting the midpoints of the sides of triangle \(ABC\). Select all true statements. 

Triangles A B C and D E F. D is the midpoint of segment A B. E is the midpoint of segment B C. F is the midpoint of segment A C. Line D E has length 2, Line E F has length 3, Line D F has line 4.  
A:

Triangle \(BDE\) is congruent to triangle \(EFC\)

B:

Triangle \(BDE\) is congruent to triangle \(DAF\)

C:

\(BD\) is congruent to \(FE\)

D:

The length of \(BC\) is 8 

E:

The length of \(BC\) is 6

(From Unit 3, Lesson 5.)