An equilateral triangle PQR has sides of length 8 cm - Leaving Cert Mathematics - Question 5 - 2022

The formula for the area of an equilateral triangle is given by:

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

In this case, the base PQ = 8 cm.

The height can be calculated using:

$height = \frac{\sqrt{3}}{2} \times side$

Substituting the side length:

$height = \frac{\sqrt{3}}{2} \times 8 = 4\sqrt{3}$

Thus, the area becomes:

$Area = \frac{1}{2} \times 8 \times 4\sqrt{3} = 16\sqrt{3} \text{ cm}^2$

We already derived the height in part (ii). Therefore, the perpendicular height from point R to base PQ is:

$h = 4\sqrt{3} \text{ cm}$

In triangle GH K, we apply the Pythagorean theorem:

$|GK|^2 = |GH|^2 + |HK|^2$

Substituting the known values:

$30^2 = 12^2 + |HK|^2$ $900 = 144 + |HK|^2$ $|HK|^2 = 900 - 144 = 756$

Taking the square root gives:

$|HK| = \sqrt{756} \approx 27.5 ext{ cm}$ (to 1 decimal place)