Question : X, Y, and Z are three equilateral triangles. The sum of the areas of X and Y is equal to the area of Z. If the side lengths of X and Y are 6 cm and 8 cm respectively, then what is the side length of Z?
Option 1: 10 cm
Option 2: 10.5 cm
Option 3: 9.5 cm
Option 4: 9 cm
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Correct Answer: 10 cm
Solution : X, Y, and Z are three equilateral triangles. The side of X = 6 cm The side of Y = 8 cm Area of a equilateral triangle = $\frac{\sqrt{3}}{4}a^2$, where $a$ = side According to the question, Area X + Area Y = Area Z $⇒ (\frac{\sqrt{3}}{4} \times 6^2) + (\frac{\sqrt{3}}{4} \times 8^2) = (\frac{\sqrt{3}}{4} \times x^2)$ $⇒ 36 + 64 = x^2$ $⇒ 100 = x^2$ $⇒ x = 10$ cm Hence, the correct answer is 10 cm.
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