Similarity in Right Triangles

1
Similarity in Right Triangles

• Lesson 8.1
• Pre-AP Geometry

2
Lesson Focus

• Right triangles have many interesting
  properties. This lesson begins with a study of
  the properties of right triangles. Algebraic
  skills with radicals are reviewed and are used
  throughout the chapter.

3
Simplifying Radicals

• Product Rule for Radicals
• For any nonnegative numbers a and b and any
  natural number index n,
• Quotient Rule for Radicals
• For any natural number index n and any real
  numbers a and b (b ? 0) where and
  are real numbers,

4
Simplifying Radicals

• Radical expressions are written in simplest terms
  when
• The index is as small as possible
• The radicand contains no factor (other than 1)
  which is the nth power of an integer or
  polynomial.
• The radicand contains no fractions.
• No radicals appear in the denominator.

5
Simplifying Radicals

• Example 1
• Example 2
• Example 3
• Example 4

6
Geometric Mean

• If a, b, and x are positive numbers with
  , then x is the geometric mean between a
  and b.
• This implies that .

7
Geometric Mean

• Example 5 Find the geometric mean of 4 and 9.
• Example 6 Find the geometric mean of 3 and 48.

8
Important

• To be successful in this chapter, you should
  spend the time to memorize the following theorem
  and corollaries.
• It is very important to your success in this
  chapter to remember these rules and know how use
  them.

9
Right Triangle Similarity Theorem

• If the altitude is drawn to the hypotenuse of a
  right triangle, then the two triangles formed are
  similar to the original triangle and to each
  other.
• ?ACB ? ?ADC ? ?CDB

10
Corollary 1

• When the altitude is drawn to the hypotenuse of
  a right triangle, the length of the altitude is
  the geometric mean between the segments of the
  hypotenuse.

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Corollary 2

• When the altitude is drawn to the hypotenuse of
  a right triangle, each leg is the geometric mean
  between the hypotenuse and the segment of the
  hypotenuse that is adjacent to that leg.

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Similarity in Right Triangles

• Example 7 If BD 16 and AD 9,
• find CD, AB, CB, and
  AC.

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Similarity in Right Triangles

• Example 8 If BD 4 and CD 2,
• find AD, AB, CB, and
  AC.

14
Written Exercises

• Problem Set 8.1A, p.288 2 - 38 (even)

15
Written Exercises

• Problem Set 8.1B, Handout 8-1