# 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 \(ABC\).

\(ABC\)

\(ACD\)

\(BCD\)

\(BDC\)

\(CAD\)

\(CBD\)

### 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 \(h\)?

### 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?

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

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

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

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.

### Problem 5

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

- What is the length of segment \(A'A\)?
- What is the length of segment \(B’B\)?

### Problem 6

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

4.5

5

7.5

10

### Problem 7

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

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

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

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

The length of \(BC\) is 8

The length of \(BC\) is 6