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Puzzle #4

An isosceles triangle where the base is divided into two equal parts by a dot and the two equal sides are divided into three equal parts by two dots. A green kite is inside the triangle where the top vertex coincides with the vertex of the triangle and the other vertices are the two top dots of the equal sides and the dot in the base of the triangle.

The area of the isosceles triangle is 108. The dots separe the sides in equal parts.

What is the area of the kite?

Solution

Let bb and hh be the base and the height of the isosceles triangle, respectively.

An isosceles triangle where the base is divided into two equal parts by a dot and the two equal sides are divided into three equal parts by two dots. A green kite is inside the triangle where the top vertex coincides with the vertex of the triangle and the other vertices are the two top dots of the equal sides and the dot in the base of the triangle. Their base and height have a double arrow and are labeled/labelled b and h, respectively.

Then,

b×h2=bh2=108\frac{b \times h}{2}=\frac{bh}{2}=108

So,

bh=216(1)bh=216\;\;\;\left(1\right)

The area of each of the two small triangles is

12b×23h2\frac{\frac{1}{2}b \times \frac{2}{3}h}{2}
13bh2\frac{\frac{1}{3}bh}{2}
16bh(2)\frac{1}{6}bh\;\;\;\left(2\right)

Substituting (1) into (2) we get:

16×216\frac{1}{6} \times 216
3636

So, the area of the kite is

108−2×36108-2\times 36
3636

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