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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