Use this formula to check your answers in the Figure it Out on page 39.

Question

Homi · Accepted Answer

Understand the Question

The area of a square with side length $s$ is given by the formula $\text{Area} = s \times s = s^2$ .
A diagonal splits a square into two congruent right-angled triangles, each having an area of $\dfrac{1}{2} s^2$ .
To check the values from the activity, calculate the theoretical area using these formulas and verify whether they match the given results.

Step 1 · State the Formula for the Area of a Square

For a square with side length $s$ : Diagram 1

\begin{aligned} \text{Area of a square} &= s \times s \\ &= s^2 \end{aligned}

Step 2 · Identify Shapes and Dimensions

The figure consists of two squares, each divided into two right-angled triangles by a diagonal cut. Diagram 2

Let:

Step 3 · Apply the Formula to Check the Answers

1. Area of the first square:

$\text{Calculated Area}_1 = s_1^2$

Compare this calculated value with the given answer $A_1$ . If $\text{Calculated Area}_1 = A_1$ , then the answer is correct.

2. Area of the second square:

$\text{Calculated Area}_2 = s_2^2$

Compare this calculated value with the given answer $A_2$ . If $\text{Calculated Area}_2 = A_2$ , then the answer is correct.

3. Area of individual triangles:

Since the diagonal divides the square into two equal halves:

$\text{Area of triangle 1} = \dfrac{1}{2} s_1^2$

Compare this with the given answer for triangle 1.

Answer

To check the answers:

First square: Calculate $s_1^2$ and compare with $A_1$ .
Second square: Calculate $s_2^2$ and compare with $A_2$ .
Triangle 1: Calculate $\dfrac{1}{2} s_1^2$ and compare with the given value.

Common Mistakes

Perimeter vs. Area Confusion: Using $4s$ instead of $s^2$ to calculate the area of the square.
Triangle Area Oversight: Forgetting that a diagonal splits a square into two equal triangles, so the area of each triangle is $\dfrac{1}{2}s^2$ rather than $s^2$ .

Question 14