Chapter 14, Problem 4NT

NOW TRY THIS

To derive the formula for area of a triangle from the formula for the area of a rectangle, cut out a triangle as in Figure 23(a); fold a perpendicular to the base with measurement b and containing point B to find the height h; and fold the altitude in half, as shown in Figure23(b). Next fold along the colored segments in the trapezoid in Figure23(c) to obtain rectangle in Figure 23(d).

a. What is the area of the rectangle in Figure 23(d)?

b. How can the formula for the area of a triangle be developed from your answer in part (a)?

To determine

(a)

To find:

The area of rectangle of Figure 23(d).

Answer to Problem 4NT

Solution:

Area of rectangle be $\left(\frac{bh}{4}\right){\text{unit}}^{\text{2}}$.

Explanation of Solution

Given:

A rectangle with dimension $\frac{b}{2}\times \frac{h}{2}$.

Approach:

$\text{Area of rectangle}=\text{length}\times \text{breadth}$.

Calculation:

$\begin{array}{rll}\text{Length of rectangle}& =& \frac{b}{2}\text{units}.\\ \text{breadth of rectangle}& =& \frac{h}{2}\text{units}.\\ \text{Area of rectangle}& =& \text{length}\times \text{breadth}\\ & =& \frac{b}{2}\times \frac{h}{2}\\ & =& \left(\frac{bh}{4}\right){\text{unit}}^{\text{2}}\end{array}$

To determine

(b)

To find:

The area of triangle from rectangle of figure 23(d).

Answer to Problem 4NT

Solution:

$\text{Area of triangle}=\frac{1}{2}\times \text{base}\times \text{height}$.

Explanation of Solution

Given:

A rectangle of dimension $\frac{b}{2}\times \frac{h}{2}$ made from two triangles of dimension $b\times h$.

Approach:

$\text{Area of triangle}=\frac{1}{2}\times \text{base}\times \text{height}$.

Calculation:

Here rectangle, Figure $24$ (d) of dimension $\frac{b}{2}\times \frac{h}{2}$ made from two triangles, Figure $24$ (a) of dimension $b\times h$.

So,

$\begin{array}{rll}2\text{area of rectangle}& =& \text{ area of triangle}.\\ 2\left(\frac{bh}{4}\right)& =& \text{area of triangle}\\ \text{area of triangle}& =& \frac{1}{2}bh\end{array}$

Thus $\text{area of triangle}=\frac{1}{2}\times \text{base}\times \text{height}$.