Exercise 19 Page 794

We are told that one diagonal of a rhombus is three times as long as the other diagonal. Therefore, if we let $d$ be the first diagonal, the second can be expressed as $3d.$ We are also told that the area is $168$ square centimeters. Let's draw a diagram to illustrate the situation.

The area of a rhombus is half the product of its diagonals. We can substitute $A=168,$ $d_1=d,$ and $d_2=3d$ into this formula and solve for $d,$ one of the diagonals. Let's do it! Note that we only kept the principal root when solving the above equation because $d$ is the length of a diagonal and it must be positive. The length of the first diagonal is $4\sqrt{7}\text{ cm}.$ Since the second diagonal is three times $d,$ we can multiply $4\sqrt{7}$ by $3$ to obtain it. We know that the lengths of the diagonals of the rhombus are $3\sqrt{7}\text{ cm}$ and $12\sqrt{7}\text{ cm}.$ We can round these numbers to the nearest tenth to get approximate values.