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Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90° HW: Special Right Triangles WS1 (side 1 only: 45-45-90) 45°- 45°- 90° 1 1 1 1 The 45-45-90 triangle is based on the square with sides of 1 unit. 45°- 45°- 90° 1 45° 1 45° 1 45° 45° 1 If we draw the diagonals we form two 45-45-90 triangles. 45°- 45°- 90° 1 45° 1 45° 1 45° 45° 1 Using the Pythagorean Theorem we can find the length of the diagonal. 45°- 45°- 90° 1 45° 1 45° 2 1 45° 45° 1 45°- 45°- 90° 45° 1 45° 1 In a 45° – 45° – 90° triangle the hypotenuse is the square root of two times as long as each leg Rule: 45°- 45°- 90° Practice 45° 4 45° SAME 4 45°- 45°- 90° Practice 45° 9 45° SAME 9 45°- 45°- 90° Practice 45° 45° SAME 45°- 45°- 90° Practice 45°- 45°- 90° Practice 45° 45° 45°- 45°- 90° Practice =3 45°- 45°- 90° Practice 45° 3 45° SAME 3 45°- 45°- 90° Practice 45° 45° 45°- 45°- 90° Practice = 11 45°- 45°- 90° Practice 45° 11 45° SAME 11 45°- 45°- 90° Practice 45° 8 45° 45°- 45°- 90° Practice 8 * = 2 Rationalize the denominator 45°- 45°- 90° Practice 45° 8 45° SAME 45°- 45°- 90° Practice 45° 4 45° 45°- 45°- 90° Practice 4 * = 2 Rationalize the denominator 45°- 45°- 90° Practice 45° 4 45° SAME 45°- 45°- 90° Practice 45° 7 45° 45°- 45°- 90° Practice 7 * Rationalize the denominator 45°- 45°- 90° Practice 45° 7 45° SAME Find the value of each variable. Write answers in simplest radical form. Find the value of each variable. Write the answers in simplest radical form. • Know the basic triangles • Set known information equal to the corresponding part of the basic triangle • Solve for the other sides 10 2 x x 5 2 y 5 2 Find the value of each variable. Write answers in simplest radical form. Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90° HW: Special Right Triangles WS1 (side 2 only: 30-60-90) 30°- 60°- 90° 2 2 60° 60° 2 The 30-60-90 triangle is based on an equilateral triangle with sides of 2 units. 30°- 60°- 90° 2 30° 30° 60° 1 2 60° 2 1 The altitude cuts the triangle into two congruent triangles. Long Leg 30°- 60°- 90° This creates 30° the 30-60-90 triangle with a hypotenuse a 60° short leg and Short Leg a long leg. 30°- 60°- 90° Practice We saw that the hypotenuse is twice the short leg. 30° 2 60° 1 We can use the Pythagorean Theorem to find the long leg. 30°- 60°- 90° Practice 30° 2 60° 1 30°- 60°- 90° 30° 2 60° 1 30° – 60° – 90° Triangle In a 30° – 60° – 90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is the square root of three times as long as the shorter leg 30°-60°-90° 30°- 60°- 90° Practice 30° 8 60° 4 The key is to find the length of the short side. Hypotenuse = short leg * 2 30°- 60°- 90° Practice 30° 10 60° 5 hyp = short leg * 2 30°- 60°- 90° Practice 30° 14 60° 7 *2 30°- 60°- 90° Practice 30° 3 60° *2 30°- 60°- 90° Practice 30° 60° *2 30°- 60°- 90° Practice 30°- 60°- 90° Practice 30° 22 60° 11 Short Leg = hyp 2 30°- 60°- 90° Practice 30° 4 60° 2 30°- 60°- 90° Practice 30° 18 60° 9 30°- 60°- 90° Practice 30° 46 60° 23 30°- 60°- 90° Practice 30° 28 60° 14 30°- 60°- 90° Practice 30° 9 60° 30°- 60°- 90° Practice 30° 12 60° hyp = Short Leg * 2 30°- 60°- 90° Practice 30° 27 60° hyp = Short Leg * 2 30°- 60°- 90° Practice 30° 20 60° hyp = Short Leg * 2 30°- 60°- 90° Practice 30° 33 60° hyp = Short Leg * 2 PRACTICE Find all the missing sides for each triangle. Solving Strategy • • • Know the basic triangles Set known information equal to the corresponding part of the basic triangle Solve for the other sides Find the value of each variable. Write answers in simplest radical form. Find the value of each variable. Write answers in simplest radical form. 12 60 24 30 12 3 60 30 5.5 5.5 5.5 2 10 10 10 2 1 60 2 30 3 10 10 5 10 60 30 8 8 2 30 60 3 60 6 30 3 3 15 15 15 2 15 15 60 8 3 4 3 30 12 8 2 2 2 18 18 9 3 18 Find the distance across the canyon. 30-60-90 b Find the length of the canyon wall (from the edge to the river). b c=b*2 c Is it more or less than a mile across the canyon? 5280 ft = 1 mile