Chapter 9, Problem 24CR

To determine

Tocalculate:The length of side AJ.

Answer to Problem 24CR

The length of side AJ is $3\sqrt{6}$ .

Explanation of Solution

Given information:

A cube with Jas the midpoint of $\overline{HF}$ .

Side $AB=6$ .

Formula used:

The below property is used:

If a point is midpoint on a line, then it divides the line into two equal parts.

Calculation:

  

  $\Delta HAF$ is anequilateral.

Each side of the triangle is $6\sqrt{2}$ because $\overline{HA}$ is the hypotenuse of a ${45}^{\circ }{45}^{\circ }{90}^{\circ }\Delta (\Delta HEA)$ , $\overline{AF}$ is the hypotenuse of a ${45}^{\circ }{45}^{\circ }{90}^{\circ }\Delta (\Delta FEA)$ , and $\overline{HF}$ is the hypotenuse of a ${45}^{\circ }{45}^{\circ }{90}^{\circ }\Delta (\Delta HEF)$

  $\Delta HAJ$ is a ${45}^{\circ }{45}^{\circ }{90}^{\circ }\Delta$ .

J is the midpoint of $\overline{HF}$

  $\begin{array}{l}HJ=\frac{1}{2}(HF)\\ HJ=\frac{1}{2}(6\sqrt{2})\\ \therefore HJ=3\sqrt{2}\\ AF=\sqrt{2}(HJ)\\ AJ=\sqrt{3}(3\sqrt{2})\\ AJ=3\sqrt{6}\end{array}$