Chapter 6, Problem 43SGR

(a)

To determine

To describe the way to determine if the trapezoids in the design are isosceles.

Explanation of Solution

Given:

The following is the given design-

Calculation:

If the legs of a trapezoid are congruent, then it is an isosceles trapezoid.

The legs of each trapezoid are part of the diagonals of the larger square.

The diagonals of a square bisect the angles of the square. Thus, each base angle of a trapezoid is of measure $\frac{{90}^{\circ }}{2}={45}^{\circ }$ . Hence, base angles of each trapezoid are congruent.

If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid. (Isosceles Trapezoid Theorem)

Isosceles Trapezoid Theorem shows that the trapezoids in the design are isosceles.

(b)

To determine

To find theperimeter of one of the trapezoids.

Answer to Problem 43SGR

  $27.3$ Inch.

Explanation of Solution

Given:

The perimeter of the larger square is $48$ inch and the perimeter of the smaller square is $16$ inch.

Calculation:

The side length of the larger square is $\frac{48}{4}=12$ inch.

Find the diagonal of the larger square $d=\sqrt{s^2+s^2}=12\sqrt{2}$ inch.

The side length of the smaller square is $\frac{16}{4}=4$ inch.

Find the diagonal of the smaller square $d=\sqrt{s^2+s^2}=4\sqrt{2}$ inch.

Thus, the length of each leg of a trapezoid is $\frac{12\sqrt{2}-4\sqrt{2}}{2}=\frac{8\sqrt{2}}{2}=4\sqrt{2}$ inch.

Find the perimeter of one trapezoid in the design.

  $\begin{array}{l}12+4+4\sqrt{2}+4\sqrt{2}\\ =14+8\sqrt{2}\\ \approx 27.3\end{array}$

Hence, the perimeter of one trapezoid is $27.3$ inch.