Accel. Geometry - Concepts Similar Figures, Right Triangles, Trigonometry

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1 Accel. Geometry - Concepts Similar Figures, Right Triangles, Trigonometry Concept 16 Ratios and Proportions (Section 7.1) Ratio: Proportion: Cross-Products Property If a b = c, then. d Properties of Proportions 1. If two ratios are equal, then their reciprocals are also equal. 2. If you exchange the means of a proportion, then you form another true proportion. 3. In a proportion, if you add the value of the ratio s denominator to its numerator, then you form another true proportion. Concept 16 Similar Polygons (Section 7.2) Similar: - Corresponding angles are - Corresponding sides are Similarity Statement: Extended Proportion: Scale Factor:

2 Concept 16 Proving Triangles Similar (Section 7.3) Postulate 7-1 Angle-Angle Similarity Postulate (AA~) If of one triangle are congruent to of another triangle, then the triangles are similar. Theorem 7-1 Side-Angle-Side Similarity Theorem (SAS~) If one angle of one triangle is congruent to one angle of another triangle and the sides including the two angles are, then the triangles are similar. Theorem 7-2 Side-Side-Side Similarity Theorem (SSS~) If the corresponding sides of two triangles are 3 B 4 Y, then the triangles are similar. A 5 C 9 12 X 15 Z Concept 17 Similarity in Right Triangles (Section 7.4) Theorem 7-3 The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are to the original triangle and to each other. Geometric Mean Example: 12 and 27 The geometric mean of two positive numbers a and b is the positive number x that satisfies a x = x b. Simplified Radical Form A square root is simplified when its radicand has no factors that are perfect squares 1) Find a perfect square that goes divides into the radicand 2) Split the radicand into two factors 3) Simplify the perfect square factor Example: Simplify each radical

3 Corollary 1 to Theorem 7-3 (Finding the Altitude) The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse. Corollary 2 to Theorem 7-3 (Finding the Leg) The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to the leg. Concept 17 Proportions in Triangles (Section 7.5) Theorem 7-4 Side-Splitter Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides. Corollary to the Side-Splitter Theorem If three parallel lines intersect two transversals, then the segments intercepted on the transversals are. Theorem 7-5 Triangle-Angle-Bisector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are to the other.

4 Concept 18 - The Pythagorean Theorem (Section 8.1) Theorem 8-1 Pythagorean Theorem If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a 2 + b 2 = c 2 Pythagorean Triple: Theorem 8-2 Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then. If c 2 = a 2 + b 2, then the triangle is right. Theorem 8-3 If c 2 > a 2 + b 2, then the triangle is. Theorem 8-4 If c 2 < a 2 + b 2, then the triangle is. Concept 19 - Special Right Triangles (Section 8.2) Rationalizing the Denominator: A fraction is not in simplest form if there is a radical in the denominator. Getting the radical out of the denominator is called rationalizing it. Step 1 Multiply the denominator by itself Step 2 Multiply the numerator by the same thing Step 3 - Simplify 8 2

5 Concept 19 Trigonometry (Section 8.3) Trigonometric Ratio: Sine of angle A (sin A) = Cosine of angle A (cos A) = Tangent of angle A (tan A) = SOH CAH TOA opposite leg hypotenuse adjacent leg A Finding Missing Side Measures 1. Decide which ratio you will need (look at the side you know already and the side you want to know) 2. Write out your equation 3. Solve the equation - if variable is on top of ratio, multiply by denominator - if variable is on bottom of ratio, multiply by denominator, then divide to isolate the variable Finding Missing Angle Measures - To find a missing angle using trigonometry, you must solve the equation using inverse sine (sin - 1 ), inverse cosine (cos -1 ), or inverse tangent (tan -1 )

6 Concept 19 Angles of Elevation and Depression (Section 8.4) Angle of Elevation: Angle of Depression: angle of depression line of sight line of sight angle of elevation