Grades 9-10 Video Solutions 2024-2024_9-10

Video Transcription

A rectangle is divided into three regions of equal area. One of the regions is an equilateral triangle with a side length of 4 cm and the other two  are trapezoids, as shown in the figure. What is the length of the shorter of the parallel sides of the trapezoids?  First, we can consider that the altitude of the triangle is 2 root 3 by using the equilateral triangle, and therefore we know that its area must be 4 root 3.  The 2 root 3 length is marked in red. Then we know that the area of the rectangle is 3 times 4 root 3, which is 12 root 3.  We know that the length of the rectangle will then be 3 root 3. Because we know that the length of the rectangle is 3 root 3, and the red portion has length  2 root 3, then this question marked side will have length root 3, and that's our answer.

Video Summary

The problem involves a rectangle divided into three regions of equal area: an equilateral triangle and two trapezoids. The triangle's side length is 4 cm, giving it an area of \(4\sqrt{3}\) using its altitude of \(2\sqrt{3}\). Thus, the rectangle's total area is \(12\sqrt{3}\). The rectangle's length is \(3\sqrt{3}\), and since the triangle's altitude occupies \(2\sqrt{3}\) of this length, the remaining portion, which is the shorter side of the trapezoids, is \(\sqrt{3}\). Therefore, the length of the shorter parallel side of the trapezoids is \(\sqrt{3}\) cm.

Keywords

rectangle

equilateral triangle

trapezoids

geometry

area