Chapter 9, Problem 20CR

To determine

Tocalculate: The length of longer diagonal.

Answer to Problem 20CR

The length of longer diagonal is $4\sqrt{3}$ .

Explanation of Solution

Given information:

The shorter diagonal of a regular diagonal is $6$ .

Formula used:

Side opposite to ${60}^{\circ }$ angle $=\sqrt{3}*shorterleg$ .

Longer diagonal $=2*shorterleg$

Calculation:

  

  $\angle X={120}^{\circ }.$

  $\Delta AXC$ is anisosceles, then $\angle XCA={30}^{\circ }$ .

Thus, $\angle ACB={90}^{\circ }$ and $\Delta ACB$ is right triangle.

  $\angle ABC$ is ${60}^{\circ }$ because diagonal $\overline{AB}$ bisects the angle.

Let x = shorter leg.

Side opposite to ${60}^{\circ }$ angle $=\sqrt{3}\ast shorterleg$ .

  $\begin{array}{l}6=\sqrt{3}x\\ \frac{6}{\sqrt{3}}=x\\ \therefore x=2\sqrt{3}\end{array}$

Hypotenuse (longer diagonal) is twice ofshorter leg.

Longer diagonal $=2\ast shorterleg$

  $\begin{array}{l}=2(2\sqrt{3})\\ =4\sqrt{3}\end{array}$

The longer diagonal is $4\sqrt{3}$ .