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Geometry - Right Angles

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    Study Guide Geometry – Right Angles 1. Altitude to the Hypotenuse Let’s explore what happens when you draw an altitude to the hypotenuse of a right triangle. This special situation creates powerful relationships using similarity and geometric means. 1.1 The Setup In right triangle △ ABC, draw altitude BD to the hypotenuse AC . Figure 1 Here’s what each part represents: • AB and BC → legs of the original right triangle • AC → hypotenuse • BD → altitude drawn to the hypotenuse • AD → segment on the hypotenuse touching leg AB • DC → segment on the hypotenuse touching leg BC 1.2 What Happens When You Draw the Altitude? The altitude creates two smaller right triangles inside the original triangle. All three triangles are similar! This leads to an important result.

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    Study Guide Theorem 62 If an altitude is drawn to the hypotenuse of a right triangle, it creates two smaller right triangles that are: • Similar to the original triangle • Similar to each other This can be proven using the AA Similarity Postulate . Figure 2 Now you can clearly see the three similar triangles. Proportions from Similarity Because the triangles are similar, corresponding sides are proportional. This gives us three important proportions. Theorem 63 If an 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 touching that leg.

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    Study Guide From similarity: This means: • BC² = AC · CD • AB² = AC · AD Each leg is the geometric mean. Theorem 64 The altitude to the hypotenuse is the geometric mean of the two segments of the hypotenuse. This means: Example 1: Solve Figure 3 Using Theorems 63 and 64, write three proportions.

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    Study Guide From Theorem 63: From Theorem 64: These represent the geometric mean relationships. Example 2: Solve Find the values of x and y in Figures 4(a) – (d). Figure 4 (a) Using Theorem 63:

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    Study Guide (b) Using Theorem 64: (c) Using Theorem 63: Using Theorem 64: (d) Use segment addition: Now apply Theorem 63:

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    Study Guide Solve: Factor: So: Since x represents a length, it cannot be negative. Then use Theorem 63 again: Substitute x = 12: Summary When an altitude is drawn to the hypotenuse of a right triangle: • Three similar triangles are formed • Each leg is the geometric mean between the hypotenuse and its touching segment • The altitude is the geometric mean between the two hypotenuse segments These geometric mean relationships make solving right triangle problems much easier once you recognize the pattern.

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    Study Guide 2. Pythagorean Theorem and Its Converse 2.1 Deriving the Pythagorean Theorem Suppose in right triangle △ ABC, CD is the altitude drawn to hypotenuse AB. Figure 1 Using the geometric mean relationships from Theorem 63: Now add the two equations: Factor out c: But from the Segment Addition Postulate: So: This is the Pythagorean Theorem .

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    Study Guide 2.2 Theorem 65 (Pythagorean Theorem) In any right triangle: Figure 2 Example 1: Finding the Hypotenuse Figure 3 Given legs 3 and 4, find x.

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    Study Guide Example 2: Solve Figure 4 2.3 Pythagorean Triples Three natural numbers ( a, b, c ) that satisfy are called a Pythagorean triple . Examples: • 3 - 4 - 5 • 5 - 12 - 13 • 8 - 15 - 17

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