9.3 Warmup Find the value of x and y. 1. 2. 3. 4 x 10 6. Are these the sides of a triangle? If yes, is the 5. acute, obtuse or right? a. 4, 4, 10 b. 9, 15,12 c. 2, 3, 4 𝑥 36 𝑦 January 18, 2019 Geometry 9.1 Similar Right Triangles
9.3 Similar Right Triangles Geometry 9.3 Similar Right Triangles 9.3 Work Sheet
Geometry 9.3 Similar Right Triangles 9.3 Essential Question How are altitudes and geometric means of right triangles related? January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Goals Know proportions in similar right triangles. Solve problems involving similar right triangles formed by altitudes on the hypotenuse. January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Means (Averages) Arithmetic mean of x & y: Geometric mean of x & y: January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Geometric Mean The geometric mean of two positive numbers a and b is the positive number x that satisfies 𝑎 𝒙 = 𝒙 𝑏 . So, 𝒙 2 =𝑎𝑏 and 𝒙= 𝑎𝑏 . January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Example 1 a. Find the geometric mean of 24 and 48. b. Find the geometric mean of 18 and 54. January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Your Turn a. Find the geometric mean of 12 and 27. b. Find the geometric mean of 16 and 18. January 18, 2019 Geometry 9.3 Similar Right Triangles
Remember AA~ for Triangles Theorem: If two angles of one triangle are congruent to two angles of a another triangle, then the triangles are similar. (AA~ Postulate) January 18, 2019 Geometry 9.3 Similar Right Triangles
Right Triangle Similarity Theorem Start with right ABC with altitude 𝐶𝐷 ( CD AB at D). C B A D January 18, 2019 Geometry 9.3 Similar Right Triangles
Right Triangle Similarity Theorem C B A D January 18, 2019 Geometry 9.3 Similar Right Triangles
Right Triangle Similarity Theorem C D B January 18, 2019 Geometry 9.3 Similar Right Triangles
Right Triangle Similarity Theorem C 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. C A C D B D B A January 18, 2019 Geometry 9.3 Similar Right Triangles
Right Triangle Similarity Theorem C ABC ~ ACD ~ CBD (AA~ Postulate) C A C D B D B A January 18, 2019 Geometry 9.3 Similar Right Triangles
Right Triangle Similarity Theorem ABC ~ ACD ~ CBD C For clarity, we name the segments. b a h B A y D x c January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles ABC ~ ACD ~ CBD C 𝑎 𝑏 = 𝒉 𝑦 = 𝑥 𝒉 ⇒ ℎ 2 =𝑥𝑦 ℎ= 𝑥𝑦 a h C A C a b c D x b B h y D B A January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles ABC ~ ACD ~ CBD C 𝒂 𝑐 = ℎ 𝑏 = 𝑥 𝒂 ⟹ 𝑎 2 =𝑥𝑐 𝑎= 𝑥𝑐 a h C A C a b c D x b B h y D B A January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles ABC ~ ACD ~ CBD C 𝒃 𝑐 = 𝑦 𝒃 = ℎ 𝑎 ⇒ 𝑏 2 =𝑦𝑐 𝑏= 𝑦𝑐 a h C A C a b c D x b B h y D B A January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Similar Triangles ABC ~ ACD ~ CBD These expressions are called geometric means. ℎ= 𝑥𝑦 𝑎= 𝑥𝑐 𝑏= 𝑦𝑐 A C D a b x y h c B January 18, 2019 Geometry 9.3 Similar Right Triangles
Theorem 9.7 Geometric Mean (Altitude) Thm The altitude drawn to the hypotenuse of a right triangle is the geometric mean of the segments on the hypotenuse. 𝒉= 𝒙𝒚 h y x January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Example 2 Find h. ℎ= 𝑥𝑦 h 6 9 4 January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Your Turn Find x. ℎ= 𝑥𝑦 4 2 8 x January 18, 2019 Geometry 9.3 Similar Right Triangles
Theorem 9.8 Geometric Mean (Leg) Thm The length of each leg of a right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. 𝒂= 𝒙𝒄 𝒃= 𝒚𝒄 b a y x c January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Example 3 Find a & b. 𝑎= 𝑥𝑐 𝑏= 𝑦𝑐 b a 10 5 15 January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Your Turn Find a & b. 𝑎= 𝑥𝑐 𝑏= 𝑦𝑐 b a 8 2 January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles All you need to know… ABC ~ ACD ~ CBD ℎ= 𝑥𝑦 𝑎= 𝑥𝑐 𝑏= 𝑦𝑐 D a b x y h c January 18, 2019 Geometry 9.3 Similar Right Triangles
Quickly Complete the Equation 6 x t 𝐲= 𝟒 × 𝟏𝟎 4 y January 18, 2019 Geometry 9.3 Similar Right Triangles
Quickly Complete the Equation 𝑥= 7×11 𝑡= 7×4 7 x t 4 y January 18, 2019 Geometry 9.3 Similar Right Triangles
Example 4: Solve for a, b, & c. 6 x 4 c January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Solution a b 6 x 4 c Begin here January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Solution a b 6 9 4 13 c January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Solution 10.82 b 6 9 4 13 January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Solution 10.82 7.21 6 9 4 13 January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles This is hard! Open your eyes! No, it isn’t. Ask: what segment do you want to find? Which others do you need to know? Which formula form? Solve the formula for the missing segment. January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles All you need to know… ABC ~ ACD ~ CBD ℎ= 𝑥𝑦 𝑎= 𝑥𝑐 𝑏= 𝑦𝑐 D a b x y h c January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles You do it. Solve for a, b, h. b a 6 h 15 21 January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Try this. Solve for d, e, & f. d e 6 f 4.5 6= 4.5 𝑓 36=4.5 𝑓 𝑓=8 January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Try this. Solve for d, e, & f. d e 6 8 4.5 12.5 d= 12.5(8 ) 𝑑= 100 𝑑=10 e= 12.5(4.5 ) 𝑑= 56.25 𝑑=7.5 January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Is this possible? 9 h 16 5 No: 9 ≠ 8.94 January 18, 2019 Geometry 9.3 Similar Right Triangles
Geometry 9.3 Similar Right Triangles Homework January 18, 2019 Geometry 9.3 Similar Right Triangles