Chapter 3.4, Problem 19E

a.

To determine

Find the length of the sides of the trapezoid.

Explanation of Solution

Given:

The perimeter of the trapezoid is 58 inches.

The average of the bases is 19 inches and longer base is twice the legs plus 7.

Shorter base = a, longer base = c and congruent legs = b

Approach:

1. $\text{Perimeter}\text{of}\text{the}\text{trapezoid}=a+2b+c$
2. $\text{Longer}\text{base}c=2b+7$

Calculation:

Consider,

  $\text{Perimeter}\text{of}\text{the}\text{trapezoid}=a+2b+c$

  $58=a+2b+c\mathrm{.......}(1)$

The average of the bases $\frac{a+c}{2}=19$

  $a+c=38\mathrm{......}(2)$

Putting the value of equation (2) in equation (1),

  $58=a+2b+c$

  $58=38+2b$

  $b=10\text{inches}$

  $\text{Longer}\text{base}c=2b+7$

Putting the value of b in above equation,

  $c=27\text{inches}$

Putting the value of c in equation (2),

  $a+c=38$

  $a+27=38$

  $a=11\text{inches}$

Therefore, the lengths of the sides of the trapezoid a, b and c are 11 inches, 10 inches and 27 inches respectively.

b.

To determine

Find the area of trapezoid.

Explanation of Solution

Given:

Shorter base = 11 inches, Longer base =27 inches and Congruent legs = 10 inches.

Approach:

1. $\text{The area of trapezoid}=\frac{a+c}{2}h$ Here, h = height of the trapezoid.
2. Find the value of h.

Calculation:

Consider, $\text{The area of trapezoid}=\frac{a+c}{2}h$

  

Height $h=\sqrt{b^2-{\left[\frac{1}{2}(c-a)\right]}^2}$

  $h=\sqrt{{10}^2-{\left[\frac{1}{2}(27-11)\right]}^2}$

  $h=\sqrt{{10}^2-8^2}$

  $h=6\text{inches}$

  $\text{The area of trapezoid}=\frac{a+c}{2}h$

  $Area=\frac{27+11}{2}\times 6$

  $Area=114{\text{inches}}^{\text{2}}$

Therefore, the area of trapezoid is $114{\text{inches}}^{\text{2}}$