Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm :

Question

Accepted Answer

To find the perimeter of the rectangle, we need to determine both the length and the width of the rectangle. We are given that the length is 40 cm and the diagonal is 41 cm. We can use the Pythagorean theorem to find the width.

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides (length and width).

Let's denote the width as $ w $.

Using the Pythagorean theorem:
$$ 40^2 + w^2 = 41^2 $$

Calculating the squares:
$$ 1600 + w^2 = 1681 $$

Subtract 1600 from both sides to solve for $ w^2 $:
$$ w^2 = 1681 - 1600 $$
$$ w^2 = 81 $$

Taking the square root of both sides:
$$ w = 9 $$

Now that we have both the length and the width, we can calculate the perimeter of the rectangle. The formula for the perimeter $ P $ of a rectangle is:
$$ P = 2 \times (\text{length} + \text{width}) $$

Substitute the known values:
$$ P = 2 \times (40 + 9) $$
$$ P = 2 \times 49 $$
$$ P = 98 $$

Therefore, the perimeter of the rectangle is 98 cm, which corresponds to Option 3.

**Explanation for why other options are incorrect:**

1. **Option 1 (90 cm):** This option is incorrect because it does not account for the correct calculation of the width using the Pythagorean theorem. The width calculated leads to a perimeter of 98 cm, not 90 cm.

2. **Option 2 (96 cm):** This option is incorrect as it also does not match the calculated perimeter using the correct width of 9 cm. The correct perimeter is 98 cm.

3. **Option 4 (108 cm):** This option is incorrect because it exceeds the calculated perimeter. The correct perimeter, based on the given dimensions and the Pythagorean theorem, is 98 cm.

In conclusion, the correct perimeter of the rectangle is 98 cm, which is why Option 3 is the correct answer.