2Objectives/DFA/HW Objectives SWBAT use properties of 45o-45o-90o & 30o-60o-90o triangles.Why? Ex. – To find the distance from home plate to 2nd on a baseball diamond.DFA – p.504 #18HW – pp (2-32 even, all)
3Side lengths of Special Right Triangles Right triangles whose angle measures are 45°-45°-90° or 30°-60°-90° are called special right triangles. The theorems that describe these relationships of side lengths of each of these special right triangles follow.
4Theorem 8.5: 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg.45°√2x45°Hypotenuse = √2 ∙ leg
5Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle Find the value of xBy the Triangle Sum Theorem, the measure of the third angle is 45°. The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg.3345°x
6Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle 3345°x45°-45°-90° Triangle TheoremSubstitute valuesSimplifyHypotenuse = √2 ∙ leg x = √2 ∙ 3 x = 3√2
7Ex. 2: Finding a leg in a 45°-45°-90° Triangle Find the value of x.Because the triangle is an isosceles right triangle, its base angles are congruent. The triangle is a 45°-45°- 90° right triangle, so the length of the hypotenuse is √2 times the length x of a leg.5xx
8Ex. 2: Finding a leg in a 45°-45°-90° Triangle Statement:Hypotenuse = √2 ∙ leg5 = √2 ∙ xReasons:45°-45°-90° Triangle TheoremSubstitute values5√2x=Divide each side by √2√2√25=xSimplify√2Multiply numerator and denominator by √2√25=x√2√25√2Simplify=x2
9Theorem 8.6: 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.60°30°√3xHypotenuse = 2 ∙ shorter legLonger leg = √3 ∙ shorter leg
10Ex. 3: Finding side lengths in a 30°-60°-90° Triangle Find the values of s and t.Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.60°30°
11Ex. 3: Side lengths in a 30°-60°-90° Triangle Statement:Longer leg = √3 ∙ shorter leg5 = √3 ∙ sReasons:30°-60°-90° Triangle TheoremSubstitute values5√3s=Divide each side by √3√3√35=sSimplify√3Multiply numerator and denominator by √3√35=s√3√35√3Simplify=s3
12Statement: Reasons: Substitute values Simplify The length t of the hypotenuse is twice the length s of the shorter leg.60°30°Statement:Hypotenuse = 2 ∙ shorter legReasons:30°-60°-90° Triangle Theorem5√3t2 ∙Substitute values=310√3Simplifyt=3
13Using Special Right Triangles in Real Life Example 4: Finding the height of a ramp.Tipping platform. A tipping platform is a ramp used to unload trucks. How high is the end of an 80 foot ramp when it is tipped by a 30° angle? By a 45° angle?
14Solution:When the angle of elevation is 30°, the height of the ramp is the length of the shorter leg of a 30°-60°-90° triangle. The length of the hypotenuse is 80 feet.80 = 2h 30°-60°-90° Triangle Theorem40 = h Divide each side by 2.When the angle of elevation is 30°, the ramp height is about 40 feet.
15Solution:When the angle of elevation is 45°, the height of the ramp is the length of a leg of a 45°-45°-90° triangle. The length of the hypotenuse is 80 feet.80 = √2 ∙ h 45°-45°-90° Triangle Theorem80=hDivide each side by √2√256.6 ≈ hUse a calculator to approximateWhen the angle of elevation is 45°, the ramp height is about 56 feet 7 inches.
16Ex. 5: Finding the distance from home plate to 2nd base on a baseball field A baseball diamond is a square with sides of 90 feet. What is the shortest distance, to the nearest tenth of a foot, between home plate and second base?