Geometry Chapter 7 7-4: SPECIAL RIGHT TRIANGLES

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Transcription:

Geometry Chapter 7 7-4: SPECIAL RIGHT TRIANGLES

Warm-Up Simplify the following. 1.) 10 30 2.) 45 5 3.) 88 8 4.) 3 27

Special Right Triangles Objective: Students will be able to use the relationships amongst the sides in special right triangles to find side lengths. Agenda 45 45 90 Triangles 30 60 90 Triangles Examples

45 45 90 Triangles Definition A 45 45 90 Triangle is an isosceles right Triangle, with 45 as the measures of both the other two angles. Leg 45 Leg 45

45 45 90 Triangles Definition A 45 45 90 Triangle is an isosceles right Triangle, with 45 as the measures of both the other two angles. Leg 45 Knowledge Connection Both Legs in this triangle are congruent. Leg 45

45 45 90 Theorem Theorem 7.8: In a 45 45 90 right triangle, the hypotenuse is 2 times as long as a leg. Hyp = Leg 2 Leg a 45 c b Leg 45

45 45 90 Examples Find the value of x.

45 45 90 Examples Find the value of x. Leg 45

45 45 90 Examples Find the value of x. 45 Leg Solution: Hyp = Leg 2 x = 12 2 x = 12 2

45 45 90 Examples Find the value of x.

45 45 90 Examples Find the value of x. Leg 45 Solution: Hyp = Leg 2 8 = x 2

45 45 90 Examples Find the value of x. Leg 45 Solution: Hyp = Leg 2 8 = x 2 x = 8 2 2 2 = 8 2 2 x = 4 2

45 45 90 Examples Find the values of x and y.

45 45 90 Examples Find the value of x and y. 45 Leg Leg

45 45 90 Examples Find the value of x and y. 45 For x: Hyp = Leg 2 2 6 = x 2 x = 2 6 2 Leg Leg x = 2 3

45 45 90 Examples Find the value of x and y. 45 For x: Hyp = Leg 2 2 6 = x 2 For y: In a 45 45 90 triangle, the Legs have the same length. Leg Leg x = 2 6 2 Therefore, y = 2 3 x = 2 3

45 45 90 Examples Find the value of x. x 8 8

45 45 90 Examples Find the value of x. x 8 Leg Leg 8

45 45 90 Examples Find the value of x. For x: Hyp = Leg 2 x = 8 2 x 8 Leg x = 8 2 Leg 8

30 60 90 Triangles Definition A 30 60 90 is a right triangle with 30 and 60 as its other angle measures. Longer Leg Shorter Leg

30 60 90 Triangles Definition A 30 60 90 is a right triangle with 30 and 60 as its other angle measures. Knowledge Connection The leg Opposite the 30 angle is called the Shorter Leg. The Leg Opposite the 60 angle is called the Longer Leg. Longer Leg Shorter Leg

30 60 90 Theorem Theorem 7-9: In a 30 60 90 right triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as a shorter leg. Hyp = S. L. 2 Longer Leg b c L. L. = S. L. 3 a Shorter Leg

30 60 90 Examples Find the values of x and y.

30 60 90 Examples Find the values of x and y. Shorter Leg Longer Leg

30 60 90 Examples Find the values of x and y. Shorter Leg Longer Leg For x: Hyp = S. L. 2 x = 6 2 x = 12

30 60 90 Examples Find the values of x and y. Shorter Leg For x: Hyp = S. L. 2 x = 6 2 Longer Leg x = 12 For y: L. L. = S. L. 3 y = 6 3 y = 6 3

30 60 90 Examples Find the values of x and y. y 20 60 x

30 60 90 Examples Find the values of x and y. Longer Leg y 20 60 x Shorter Leg

30 60 90 Examples Find the values of x and y. Longer Leg For x: y 20 60 x Shorter Leg Hyp = S. L. 2 20 = 2x x = 10

30 60 90 Examples Find the values of x and y. Longer Leg For x: y 20 60 x Shorter Leg Hyp = S. L. 2 20 = 2x x = 10 For y: L. L. = S. L. 3 y = 10 3 y = 10 3

30 60 90 Examples Find the values of x and y.

30 60 90 Examples Find the values of x and y. Shorter Leg Longer Leg

30 60 90 Examples Find the values of x and y. Shorter Leg Longer Leg For x: L. L. = S. L. 3 8 = x 3 x = 8 3 x = 8 3 3 3 = 8 3 3

30 60 90 Examples Find the values of x and y. Shorter Leg Longer Leg For x: L. L. = S. L. 3 For y: Hyp = S. L. 2 8 = x 3 x = 8 3 y = x 2 y = 2 8 3 3 x = 8 3 3 3 = 8 3 3 y = 16 3 3

30 60 90 Examples Find the values of x and y. 6 x 6 3 3

30 60 90 Examples Find the values of x and y. 6 x 6 Longer Leg 3 3 Shorter Leg

30 60 90 Examples Find the values of x and y. For x: 6 x Longer Leg 6 L. L. = S. L. 3 x = 3 3 x = 3 3 3 3 Shorter Leg

Final Practice: Both Triangles Find the values of the variables in the given diagram. For u: Hyp = Leg 2 8 2 = u 2 u = 8 2 2 For v: In a 45 45 90 triangle, the Legs have the same length. Therefore, v = 8 u = 8

Final Practice: Both Triangles Find the values of the variables in the given diagram. m For m: For n: 10 45 n Hyp = Leg 2 10 = m 2 m = 10 2 In a 45 45 90 triangle, the Legs have the same length. Therefore, n = 5 m = 5

Final Practice: Both Triangles Find the values of the variables in the given diagram. For b: In a 45 45 90 triangle, the Legs have the same length. Therefore, b = 2 2 For a: Hyp = Leg 2 a = 2 2 2 a = 2(2) a = 4

Final Practice: Both Triangles Find the values of the variables in the given diagram. For u: Hyp = S. L. 2 u = 2 2 u = 4 For v: L. L. = S. L. 3 y = 2 3 y = 2 3

Final Practice: Both Triangles Find the values of the variables in the given diagram. For y: Hyp = S. L. 2 8 5 = 2y y = 4 5 For y: L. L. = S. L. 3 y = 4 5 3 y = 4 15

Final Practice: Both Triangles Find the values of the variables in the given diagram. For b: L. L. = S. L. 3 11 3 = b 3 b = 11 3 3 b = 11 For a: Hyp = S. L. 2 a = 11 2 a = 22

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