Math Quiz - Trigonometry Report a question What's wrong with this question? You cannot submit an empty report. Please add some details. Math Quiz - Trigonometry Covering: trigonometry, which will be on trigonometric functions and ratios, Pythagorean theorem, and special right triangles (30-60-90 and 45-45-90). 1 / 5 1. A 45-45-90 triangle has a side length of 8 cm opposite one of the 45Â° angles. Find the length of the hypotenuse. a) 8 cm b) 12 cm c) 8âˆš2 cm d) 16âˆš2 cm In a 45-45-90 triangle, the hypotenuse is âˆš2 times the length of each leg (sides opposite the 45Â° angles). We are given the length of one leg (opposite a 45Â° angle) as 8 cm. To find the length of the hypotenuse (let's call it "c"), we can use the following equation: c = 8âˆš2 cm 2 / 5 2. Given that Î”ABC is a right triangle with âˆ BCA = 60Â° and AB = 8, where the triangle is inscribed in a circle with its hypotenuse serving as the diameter, determine the area of the circle. a) 27Ï€ b) 44.5Ï€ c) 64Ï€ d) 40âˆš3Ï€ Since Î”ABC is a right triangle with âˆ BCA = 60Â° and AB = 8, we can find the length of the hypotenuse AC using the Pythagorean theorem:ACÂ² = ABÂ² + BCÂ² ACÂ² = 8Â² + (8 * tan(60Â°))Â² ACÂ² = 64 + 64 * (âˆš3)Â² ACÂ² = 64 + 64 * 3 ACÂ² = 64 + 192 ACÂ² = 256Now, find the length of the hypotenuse AC:AC = âˆš256 = 16 cmSince the hypotenuse serves as the diameter of the circle, the radius R is half the length of the hypotenuse:R = AC / 2 = 16 / 2 = 8 cmFinally, find the area of the circle using the formula:Area = Ï€ * RÂ²In this case, the area of the circle is:Area = Ï€ * (8)Â² = 64Ï€ square centimeters 3 / 5 3. In a right triangle JKL, angle J = 90Â° and angle K = 60Â°. The perimeter of triangle JKL is 30 cm. Find the length of side KL. a) 5 cm b) 10 cm c) 15 cm d) 20 cm Since triangle JKL is a right triangle with angle K = 60Â°, angle L must be 30Â° (because angle J is a right angle). The triangle JKL is, therefore, a 30-60-90 triangle. In a 30-60-90 triangle, the hypotenuse (JL) is twice the length of the side opposite the 30Â° angle (KL), and the side JK (opposite the 60Â° angle) has a length that is âˆš3 times the side opposite the 30Â° angle (KL). Let "x" be the length of side KL, then JL is 2x and JK is xâˆš3. Perimeter = KL + JK + JL 30 cm = x + xâˆš3 + 2x Now we can solve for x: x(1 + âˆš3 + 2) = 30 cm x â‰ˆ 5 cm The length of side KL is approximately 5 cm. 4 / 5 4. In a right triangle LMN, angle L = 90Â° and angle M = 45Â°. If the length of LN is 9 cm, find the length of side MN. a) 3âˆš2 cm b) 6âˆš2 cm c) 9 cm d) 18 cm Since triangle LMN is a right triangle with angle M = 45Â°, angle N must be 45Â° as well (because angle L is a right angle). The triangle LMN is, therefore, a 45-45-90 triangle. In a 45-45-90 triangle, the sides opposite both of the 45Â° angles are congruent, and the hypotenuse is âˆš2 times the length of each leg. In this case, the side LN (opposite angle N) is 9 cm. Since the triangle is a 45-45-90 triangle, the side opposite angle M (MN) must also be 9 cm. 5 / 5 5. Which of the following is NOT a Pythagorean triple? a) (3, 4, 5) b) (5, 12, 13) c) (7, 24, 25) d) (8, 15, 18) A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the Pythagorean theorem, aÂ² + bÂ² = cÂ². First, we will rationalize the first option. (3,4,5): (3 cm)Â² + (4 cm)Â² = (5 cm)Â²?; 9 + 16 = 25; 25 = 25. Next, (5,12,13): (5 cm)Â² + (12 cm)Â² = (13 cm)Â²?; 25 + 144 = 169; 169 = 169. Now for (7,24,25): (7 cm)Â² + (24 cm)Â² = (25 cm)Â²?; 49 + 576 = 625; 625 = 625. Last, (8,15,18): (8 cm)Â² + (15 cm)Â² = (18 cm)Â²?; 64 + 225 = 289; 289 â‰ 324. The Pythagorean triple that doesn't meet the condition is (8, 15, 18). Your score is 0% Restart Quiz