What fraction of the total area of the rectangle is the area of the blue region?

Question

Homi · Accepted Answer

We can find the area of the blue triangles by using their bases (sides of the rectangle) and their heights (perpendicular distances from the central point).

Step 1 — Calculate total area of rectangle

Let the length of the rectangle be $L$ . Let the breadth (width) of the rectangle be $W$ . The total area of the rectangle is found by multiplying its length and breadth.

$\text{Area of rectangle} = L \times W$

$\boxed{L \times W \text{ square units}}$

Diagram 1

Step 2 — Identify blue triangles and their dimensions

The blue region consists of two triangles. Let the point where the triangles meet inside the rectangle be P. The top blue triangle has vertices A, B, and P. Its base is AB. The bottom blue triangle has vertices C, D, and P. Its base is CD. The length of base AB is $L$ . The length of base CD is $L$ .

Let $h_1$ be the perpendicular distance from point P to side AB. Let $h_2$ be the perpendicular distance from point P to side CD. The sum of these two heights is equal to the total breadth of the rectangle. So, we have $h_1 + h_2 = W$ .

Diagram 2

Step 3 — Calculate area of blue region

The area of any triangle is half times its base times its height. Let us find the area of the top blue triangle (APB).

$\text{Area of triangle APB} = \frac{1}{2} \times \text{base AB} \times h_1$

$= \frac{1}{2} \times L \times h_1$

Let us find the area of the bottom blue triangle (CPD).

$\text{Area of triangle CPD} = \frac{1}{2} \times \text{base CD} \times h_2$

$= \frac{1}{2} \times L \times h_2$

The total area of the blue region is the sum of these two areas.

$\text{Area of blue region} = \text{Area of APB} + \text{Area of CPD}$

$= \left(\frac{1}{2} \times L \times h_1\right) + \left(\frac{1}{2} \times L \times h_2\right)$

We can take out the common factor $\frac{1}{2} \times L$ .

$= \frac{1}{2} \times L \times (h_1 + h_2)$

From Step 2, we know that $h_1 + h_2 = W$ .

$= \frac{1}{2} \times L \times W$

$\boxed{\frac{1}{2} \times L \times W \text{ square units}}$

Step 4 — Find the fraction of the total area

We need to find what fraction of the total area is covered by the blue region. This is found by dividing the area of the blue region by the total area of the rectangle.

$\text{Fraction} = \frac{\text{Area of blue region}}{\text{Area of rectangle}}$

$= \frac{\frac{1}{2} \times L \times W}{L \times W}$

We can cancel out $L \times W$ from the numerator and the denominator.

$= \frac{1}{2}$

$\boxed{\frac{1}{2}}$

Answer

(i) The fraction of the total area of the rectangle that is the area of the blue region is 1/2.

Question 18

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