Chapter 11, Problem 21RP

Solution

To Calculate: Area and the perimeter of an isosceles trapezoid.

The perimeter and area of an isosceles trapezoid are $50$ unitand $77\sqrt{3}{\text{ units}}^2$ .

Given information:

Lower base of trapezoid = $18$

Upper base = $4$

Upper base angle = ${120}^{\circ }$

Formula used:

Here, we are using the formula:

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

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

Calculation:

  

We know that,

tan $\theta =\frac{P}{B}$

(Where, “ P ” is perpendicular and “ B ” is base)

In $\Delta BFD$

tan ${30}^{\circ }=\frac{BF}{FD}$

  $\frac{1}{\sqrt{3}}=\frac{7}{FD}$

  $FD=7\sqrt{3}$

So, “ B ” = $7\sqrt{3}$

And, Sin $\theta =\frac{P}{H}$

(Where, “ P ” is perpendicular and “ H ” is hypotenuse)

In $\Delta BFD$

Sin ${30}^{\circ }=\frac{FD}{BD}$

  $\frac{1}{2}=\frac{7}{BD}$

  $BD=14$

So, “ H ” = $14$

Perimeter of trapezoid = Sum of all sides

➔AB+AC+CD+BD

➔$4+14+18+14$

➔$50$

Now, area of trapezoid = $\frac{1}{2}\left(a+b\right)\times h$

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

➔$\frac{1}{2}\left(4+18\right)\times 7\sqrt{3}$

➔$77\sqrt{3}$

Conclusion: Hence, the perimeter and area of an isosceles trapezoid are $50$ unitand $77\sqrt{3}{\text{ units}}^2$