Similarity in Right Triangles Skill 38

Objective HSG-SRT.5/GPE.5: Students are responsible for finding and using the relationships in similar right triangles.

Definition For any two positive numbers a and b, the geometric mean of a and b is the positive x such that 𝒂 𝒙 = 𝒙 𝒃 .

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 ∆𝑨𝑩𝑪 ~ ∆𝑨𝑪𝑫 ~ ∆𝑪𝑩𝑫

𝑨𝑫 𝑪𝑫 = 𝑪𝑫 𝑫𝑩 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 𝑨𝑫 𝑪𝑫 = 𝑪𝑫 𝑫𝑩

𝑨𝑩 𝑨𝑪 = 𝑨𝑪 𝑨𝑫 𝒂𝒏𝒅 𝑨𝑩 𝑪𝑩 = 𝑪𝑩 𝑫𝑩 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 𝑨𝑩 𝑨𝑪 = 𝑨𝑪 𝑨𝑫 𝒂𝒏𝒅 𝑨𝑩 𝑪𝑩 = 𝑪𝑩 𝑫𝑩

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 ∆𝑿𝒀𝒁 ~ ∆𝒀𝑾𝒁 ~ ∆𝑿𝑾𝒀

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 ∆𝑷𝑸𝑹 ~ ∆𝑺𝑸𝑷 ~ ∆𝑺𝑷𝑹

Example 2; Finding Geometric Mean a) What is the geometric mean of 6 and 15? 𝟔 𝒙 = 𝒙 𝟏𝟓 𝟗𝟎 𝒙 𝟐 =𝟔 𝟏𝟓 𝟗 𝟏𝟎 𝒙 𝟐 =𝟗𝟎 𝟑 𝟏𝟎 𝒙=± 𝟗𝟎 𝒙=𝟑 𝟏𝟎

Example 2; Finding Geometric Mean b) What is the geometric mean of 7 and 18? 𝟕 𝒙 = 𝒙 𝟏𝟖 𝟏𝟐𝟔 𝒙 𝟐 =𝟕 𝟏𝟖 𝟗 𝟏𝟒 𝒙 𝟐 =𝟏𝟐𝟔 𝟑 𝟏𝟒 𝒙=± 𝟏𝟐𝟔 𝒙=𝟑 𝟏𝟒

Example 3; Using the Corollaries to Theorem 62 a) Find the values of w and x. 𝟒 𝒘 = 𝒘 𝟒+𝟏𝟐 𝟒 𝒙 = 𝒙 𝟏𝟐 w x 12 4 𝒘 𝟐 =𝟒 𝟏𝟔 𝒙 𝟐 =𝟒 𝟏𝟐 𝒘 𝟐 =𝟔𝟒 𝒙 𝟐 =𝟒𝟖 𝒘=± 𝟔𝟒 𝒙=± 𝟒𝟖 𝒘=𝟖 𝒙=𝟒 𝟑

Example 3; Using the Corollaries to Theorem 62 y z 5 4 b) Find the values of y and z. 𝟒 𝒚 = 𝒚 𝟓 𝟒 𝒛 = 𝒛 𝟗 𝒚 𝟐 =𝟒 𝟓 𝒛 𝟐 =𝟒 𝟗 𝒚 𝟐 =𝟐𝟎 𝒛 𝟐 =𝟑𝟔 𝒚=± 𝟐𝟎 𝒛=± 𝟑𝟔 𝒚=𝟐 𝟓 𝒛=𝟔

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? 𝒙+𝟗 𝟐𝟎 = 𝟐𝟎 𝒙 A x 9 20 in. D B C 𝒙 𝒙+𝟗 =𝟒𝟎𝟎 𝒙 𝟐 +𝟗𝒙=𝟒𝟎𝟎 𝒙 𝟐 +𝟗𝒙−𝟒𝟎𝟎=𝟎 𝒙+𝟐𝟓 𝒙−𝟏𝟔 =𝟎 Distance A to D is 16 inches 𝒙+𝟐𝟓=𝟎 𝒙−𝟏𝟔=𝟎 𝒙=−𝟐𝟓 𝒙=𝟏𝟔

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? 𝟗 𝒚 = 𝒚 𝟏𝟔 A 16 9 20 in. D B C 𝒚 𝟐 =𝟏𝟒𝟒 y 𝒚=± 𝟏𝟒𝟒 𝒚=𝟏𝟐 Distance D to B is 12 inches

#38: Similarity in Right Triangles Questions? Summarize Notes Homework Video Quiz