Chapter 11.6, Problem 39HP

To determine

To describe: How to formula for the area of the trapezoid is related to the formula of area of a triangle.

Answer to Problem 39HP

Area of trapezoid is the sum of total area of three triangles forming inside it, having same heights as of trapezoid.

Explanation of Solution

Given information: Formula of area of a trapezoid and the formula of area of the triangle.

Formula/Concept used:

Area of a trapezoid $=\frac{1}{2}\times \left(\text{sum of its parallel bases}\right)\times \text{height of the trapezoid}$

And Area of a triangle $=\frac{1}{2}\times \text{base}\times h\text{eight of the triangle}$

Now to show that area of trapezoid is actually the sum of areas of three triangles, forming inside it.

Calculation: Let ABCD be the trapezoid, divided into three triangles ADH, AHB and BHC, with having same heights h , as shown below :

Area of a triangle ADH $=\frac{1}{2}\times DH\times h$

Area of a triangle AHB $=\frac{1}{2}\times AB\times h$

Area of a triangle BHC $=\frac{1}{2}\times HC\times h$

On adding all above areas

Area of a triangle ADH+ Area of a triangle AHB+ Area of a triangle BHC

  $=\frac{1}{2}\times DH\times h+\frac{1}{2}\times AB\times h\frac{1}{2}\times HC\times h$

  $\begin{array}{l}=\frac{1}{2}\times (DH+AB+HC)\times h\\ ==\frac{1}{2}\times (DC+AB)\times h\end{array}$

  $=\frac{1}{2}\times \left(\text{sum of its parallel bases}\right)\times \text{height of the trapezoid}$

= Area of trapezoid ABCD

Conclusion: By above calculation, it is cleared, that area of the trapezoid is actually the sum of areas of three triangle forming inside it, having same heights as of trapezoid.