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Similarity in Right Triangles
Skill 38
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Objective HSG-SRT.5/GPE.5: Students are responsible for finding and using the relationships in similar right triangles.
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Definition For any two positive numbers a and b, the geometric mean of a and b is the positive x such that π π = π π .
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Thm. 68: Altitude Right Triangle-Similar Triangle
The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other. A D B C βπ¨π©πͺ ~ βπ¨πͺπ« ~ βπͺπ©π«
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π¨π« πͺπ« = πͺπ« π«π© Corollary 1 to Theorem 68
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. A D B C π¨π« πͺπ« = πͺπ« π«π©
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π¨π© π¨πͺ = π¨πͺ π¨π« πππ
π¨π© πͺπ© = πͺπ© π«π© Corollary 2 to Theorem 68
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. A D B C π¨π© π¨πͺ = π¨πͺ π¨π« πππ
π¨π© πͺπ© = πͺπ© π«π©
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Example 1; Identifying Similar Triangles
a) What similarity statement can you write relating the three triangles in the diagram? W X Y Z W X Y W Z Y βπΏππ ~ βππΎπ ~ βπΏπΎπ
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Example 1; Identifying Similar Triangles
b) What similarity statement can you write relating the three triangles? Q S R P S Q P S R P βπ·πΈπΉ ~ βπΊπΈπ· ~ βπΊπ·πΉ
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Example 2; Finding Geometric Mean
a) What is the geometric mean of 6 and 15? π π = π ππ ππ π π =π ππ π ππ π π =ππ π ππ π=Β± ππ π=π ππ
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Example 2; Finding Geometric Mean
b) What is the geometric mean of 7 and 18? π π = π ππ πππ π π =π ππ π ππ π π =πππ π ππ π=Β± πππ π=π ππ
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Example 3; Using the Corollaries to Theorem 62
a) Find the values of w and x. π π = π π+ππ π π = π ππ w x 12 4 π π =π ππ π π =π ππ π π =ππ π π =ππ π=Β± ππ π=Β± ππ π=π π=π π
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Example 3; Using the Corollaries to Theorem 62
y z 5 4 b) Find the values of y and z. π π = π π π π = π π π π =π π π π =π π π π =ππ π π =ππ π=Β± ππ π=Β± ππ π=π π π=π
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Example 4; Finding Distance
a) You are preparing for a robotics competition using the set up to the left. You program the robot to move from A to D and to pick up the plastic bottle placed at D. How far does the robot travel from A to D? π+π ππ = ππ π A x 9 20 in. D B C π π+π =πππ π π +ππ=πππ π π +ππβπππ=π π+ππ πβππ =π Distance A to D is 16 inches π+ππ=π πβππ=π π=βππ π=ππ
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Example 4; Finding Distance
b) From point D, the robot must turn right an move to point B to put the bottle in the recycling bin. How far does the robot travel from D to B? π π = π ππ A 16 9 20 in. D B C π π =πππ y π=Β± πππ π=ππ Distance D to B is 12 inches
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#38: Similarity in Right Triangles
Questions? Summarize Notes Homework Video Quiz
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