Chapter 11.3, Problem 19PSC

a

To determine

To prove: Area of trapezoid using a diagonal and two triangles.

Explanation of Solution

Given information : Trapezoid divided into two triangles by diagonal.

Formula used : Area of triangle = $\frac{1}{2}\times b\times h$

Prove: Let $b_1$ and $b_2$ be the bases of triangle and $h$ be the height.

Area of trapezoid = Area of Triangle 1 + Area of Triangle 2

Now, area of triangle 1 = $\frac{1}{2}\times b_1\times h$

Similarly, area of triangle 2 = $\frac{1}{2}\times b_2\times h$

  $\therefore$ Area of trapezoid = $\frac{1}{2}\times b_1\times h+\frac{1}{2}\times b_2\times h$

  $\Rightarrow \frac{1}{2}\times h\times (b_1+b_2)$ = Area of trapezoid

  

b

To determine

To prove: Area of trapezoid using altitudes, rectangles and triangles.

Explanation of Solution

Given information : Trapezoid divided into two triangles and one rectangle

Formula used : Area of triangle = $\frac{1}{2}\times b\times h$ and Area of rectangle = $l\times b$

Prove: Let $a$ and $b$ be the bases of triangle and $c$ be the length of rectangle.

Area of trapezoid = Area of Triangle 1 + Area of rectangle + Area of Triangle 2

Now, area of triangle 1 = $\frac{1}{2}\times a\times h$

Similarly, area of triangle 2 = $\frac{1}{2}\times b\times h$

Here, breadth of rectangle is same as height of triangle,

  $\Rightarrow$ Area of rectangle = $c\times h$

  $\therefore$ Area of trapezoid = $\frac{1}{2}\times a\times h+c\times h+\frac{1}{2}\times b\times h$

  $\begin{array}{l}\Rightarrow \frac{1}{2}\times a\times h+\frac{1}{2}\times 2(c\times h)+\frac{1}{2}\times b\times h\\ \Rightarrow \frac{1}{2}\times h(a+2c+b)\end{array}$

  $\Rightarrow \frac{1}{2}\times h\times (b_1+b_2)$ = Area of trapezoid