The diagram shows a square of side length 2k cm - Leaving Cert Mathematics - Question b - 2017

Given that the triangle has side lengths of 20 cm and hypotenuse of 2k cm, we can use the Pythagorean theorem to find the relationship between the sides:

$(2k)^2 = 20^2 + 20^2$

This simplifies to:

$4k^2 = 400$

Dividing both sides by 4:

$k^2 = 100$

Taking the square root gives:

$k = 10$

First, calculate the area of the square:

$\text{Area of square} = 4k^2 = 4(10)^2 = 400 \text{ cm}^2$

Now calculate the area of the triangle:

• The base is 20 cm.
• The height can be calculated as follows:

Using the formula for the area of a triangle:

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

To find the height, we can apply the formula:

Using the Pythagorean theorem on one half of the isosceles triangle:

$h^2 + 10^2 = 20^2 \implies h^2 + 100 = 400 \implies h^2 = 300 \implies h = 10\sqrt{3}$

Thus, the area of the triangle can be expressed as:

$\text{Area of Triangle} = \frac{1}{2} \times 20 \times 10\sqrt{3} = 100\sqrt{3} \approx 173.21 \text{ cm}^2$

Finally, subtract the area of the triangle from the area of the square to find the area of the shaded section:

$\text{Area of shaded section} = 400 - 173.21 \approx 226.79 \text{ cm}^2$