Chapter 11.3, Problem 16PSC

To determine

To Calculate: The height and the perimeter of the trapezoid.

Answer to Problem 16PSC

The height and the perimeter of the trapezoid are $14$ and $70$ respectively.

Explanation of Solution

Given information:

The sides of the trapezoid are in the ratio $2:5:8:5$

The area is $245$

Formula used:

Here, we are using the formula

Area of trapezoid = $\frac{1}{2}\left(a+b\right)h$

(here, “ a “is upper base, “ b ” is lower base and “ h ” is height)

Calculation:

  

Here, let the side of trapezoid be x

  $2x,5x,8x$ and $5x$

Area of trapezoid = $\frac{1}{2}\left(a+b\right)h$

(where, “ a ” is upper base, “ b ” is lower base and “ h ” is height)

And, area of trapezoid = $\frac{1}{2}\left(5x+5x\right)\times h$

  $245=\frac{1}{2}\left(10x\right)\times h$

  $xh=49$ ........ $\left(1\right)$

Here, In $\Delta ADE$

  ${\left(hypotenuse\right)}^2={\left(perpendicular\right)}^2+{\left(base\right)}^2$

  ${\left(AD\right)}^2={\left(AE\right)}^2+{\left(DE\right)}^2$

  ${\left(5x\right)}^2={\left(AE\right)}^2+{\left(3x\right)}^2$

  $A{E}^2=25x^2-9x^2$

  $AE"h"=4x$

Now,

We know that $xh=49$

  $x\times 4x=49$

  $x=\frac{7}{2}$

Now, from equation $\left(1\right)$

  $xh=49$

  $\frac{7}{2}\times h=49$

  $h=14$

Now, Perimeter of trapezoid = $2x+5x+8x+5x$ (perimeter of trapezoid = sum of all sides)

→ $2\times \frac{7}{2}+5\times \frac{7}{2}+8\times \frac{7}{2}+5\times \frac{7}{2}$

→ $7+\frac{35}{2}+28+\frac{35}{2}$

→ $70$

Conclusion: Hence,the height and the perimeter of the trapezoid are $14$ and $70$ respectively.