Chapter 11, Problem 37RP

To determine

To calculate:The areaof a trapezoidWXYZ with diagonals WY , XZ as 30 each and height WU as 18.

Answer to Problem 37RP

The area of a trapezoid WXYZ is $432$ .

Explanation of Solution

Given information:

In trapezoidWXYZ ,

DiagonalWY=XZ = 30,

Side WU = 18.

Formula used:

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Area of rectangle: $A=l\ast w$

l = length of rectangle

w= width of rectangle

Calculation:

  

If the diagonals of trapezoid are congruent, then it is isosceles.

ΔWUZ and ΔXAY are congruent by HL congruence rule.

Area of the shaded region = Area of region I.

Side WA can be calculated by using Pythagoras Theorem.

In right angled triangle WAY , we get

  ${\left(WY\right)}^2={\left(WA\right)}^2+{\left(AY\right)}^2$

  $\begin{array}{l}{\left(WA\right)}^2={\left(WY\right)}^2-{\left(AY\right)}^2\\ {\left(WA\right)}^2={\left(30\right)}^2-{\left(18\right)}^2\\ {\left(WA\right)}^2=900-324\\ {\left(WA\right)}^2=576\\ WA=\sqrt{576}=24\end{array}$

Area of trapezoid WXYZ = Area of rectangle WUYA

  $\begin{array}{l}A(\square WUYA)=UY\ast WU\\ A(\square WUYA)=UY\ast WU\\ UY=WA=24\\ A(\square WUYA)=24\ast 18=432\end{array}$

Area of trapezoid WXYZ $=432$