3420.61 – Two Triangles and a Square

In the diagram below two triangles and a square are drawn, but not to scale. The triangles I and III are equilateral of area 32332\sqrt{3} and 83 8\sqrt{3} square inches, respectively. The square II has area 3232 square inches.

If the segment ADAD is decreased by 12.5 percent while lengths ABAB and CDCD are unchanged, what is the percent change in the area of the square?

3420_61_e6d29dcb1d.png

Solution

The area of an equilateral triangle of side ss is s234\frac{s^2\sqrt{3}}{4}. Referring to the lengths in the diagram below, this implies that

For I: x234=323⇝x24=32⇝x2=128⇝x=82,For II: y2=32⇝y=42,For III: z234=83⇝z2=32⇝z=42. \begin{aligned} \text{For I: } \frac{x^2\sqrt{3}}{4} = 32\sqrt{3} &\leadsto& \frac{x^2}{4} = 32 \leadsto x^2 = 128 \leadsto x = 8\sqrt{2},\\ \text{For II: } y^2 = 32 &\leadsto& y = 4\sqrt{2}, \\ \text{For III: } \frac{z^2\sqrt{3}}{4} = 8\sqrt{3} &\leadsto& z^2 = 32 \leadsto z = 4\sqrt{2}. \end{aligned}

So AD=82+42+42=162.AD = 8\sqrt{2} + 4\sqrt{2} + 4\sqrt{2} = 16\sqrt{2}. If ADAD decreases by 12.5 percent, that is by one-eighth, then it becomes 14214\sqrt{2}. Lengths AB and BDAB \text{ and } BD are unchanged so BCBC absorbs the whole decrease of 222\sqrt{2} and shrinks from 424\sqrt{2} to 222\sqrt{2}. The area of the square is now (22)2=8,\left( 2\sqrt{2} \right)^2 = 8, down from 3232, a loss of 75 percent.

3420_61_solution_09ae6cab98.png