ABC is a triangle with ∠BAC = 120°, and D and E are points on side BC such that DAE is an equilateral triangle. If BD = 8 cm and EC = 32 cm, what is the length (in cm) of DE?
Question
ABC is a triangle with ∠BAC = 120°, and D and E are points on side BC such that DAE is an equilateral triangle. If BD = 8 cm and EC = 32 cm, what is the length (in cm) of DE?
Solution
To solve this problem, we need to use the properties of equilateral triangles and the Law of Cosines.
Step 1: Identify the given information. We know that ∠BAC = 120°, BD = 8 cm, and EC = 32 cm. We also know that triangle DAE is an equilateral triangle, so AD = AE = DE.
Step 2: Use the Law of Cosines to find the length of AD. The Law of Cosines states that c² = a² + b² - 2ab cos(C), where a, b, and c are the sides of a triangle and C is the angle opposite side c. In triangle ABC, a = BD = 8 cm, b = EC = 32 cm, and C = ∠BAC = 120°. Plugging these values into the Law of Cosines gives us AD² = 8² + 32² - 2832*cos(120°) = 64 + 1024 - 512 = 576, so AD = √576 = 24 cm.
Step 3: Since triangle DAE is an equilateral triangle, DE = AD = 24 cm. So, the length of DE is 24 cm.
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