A family is laying new turf and paving for their back garden, which is shaped like a rectangle with an isosceles triangle section added on top, as shown in ...
Question 1 Report
A family is laying new turf and paving for their back garden, which is shaped like a rectangle with an isosceles triangle section added on top, as shown in the diagram. All measurements are in metres and the shape is symmetrical.
- Work out the area of the garden. (3)
- Work out the perimeter of the garden, giving your answer correct to 1 decimal place. (2)
Answer Details
This composite shape is a rectangle with an isosceles triangle attached along one side, so its area is the sum of the rectangle's area and the triangle's area, and its perimeter requires the two equal slant sides of the triangle to be found using Pythagoras' theorem.
(a) Area of the rectangular part, 12 m by 6 m:
\[ 12 \times 6 = 72 \text{ m}^2 \]Area of the triangular part, base 12 m and height 4 m:
\[ \frac{1}{2} \times 12 \times 4 = 24 \text{ m}^2 \]Total area:
\[ 72 + 24 = 96 \text{ m}^2 \][3]
(b) Each slant side of the triangle is the hypotenuse of a right-angled triangle with a horizontal leg of \(12 \div 2 = 6\) m and a vertical leg of 4 m:
\[ \text{slant side} = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} = 7.2111\ldots \text{ m} \]The perimeter is the two rectangle sides of 6 m, the base of 12 m, and the two slant sides:
\[ 12 + 6 + 6 + 7.2111 + 7.2111 = 38.4 \text{ m (1 d.p.)} \][2]
Because the shape is symmetrical, only one slant length needs to be calculated using Pythagoras' theorem; the other slant side is identical by symmetry, which halves the working needed.
Download The App On Google Playstore
Everything you need to excel in your exams
