Question

the diagram shows a shapes stencil.
the stencil is made from a square of plastic, with the four shapes cut out.
the shaded area is 126cm².
what is the height, h, of the triangle?
tip
remember:
the formula for the area of a trapezium is:
area = \\(\frac{1}{2}(a + b) \times h\\)
the formula for the area of a parallelogram is:
area = base × perpendicular height

Explanation:

Step1: Calculate area of square

The square has side length 15 cm (from the diagram). Area of square is \( \text{side}^2 = 15\times15 = 225 \, \text{cm}^2 \).

Step2: Calculate area of rectangle

Rectangle dimensions: 7 cm (length) and 4 cm (width). Area \( = 7\times4 = 28 \, \text{cm}^2 \).

Step3: Calculate area of trapezium

Trapezium has \( a = 4 \, \text{cm} \), \( b = 6 \, \text{cm} \), \( h = 5 \, \text{cm} \). Area \( = \frac{1}{2}(4 + 6)\times5 = \frac{1}{2}\times10\times5 = 25 \, \text{cm}^2 \).

Step4: Calculate area of parallelogram

Parallelogram: base = 6 cm, height = 4 cm. Area \( = 6\times4 = 24 \, \text{cm}^2 \).

Step5: Calculate area of triangle (unknown height \( h \))

Triangle: base = 5 cm, height = \( h \). Area \( = \frac{1}{2}\times5\times h = \frac{5h}{2} \, \text{cm}^2 \).

Step6: Sum of cut - out areas

Sum of areas of rectangle, trapezium, parallelogram and triangle: \( 28+25 + 24+\frac{5h}{2}=77+\frac{5h}{2} \).

Step7: Set up equation for shaded area

Shaded area = Area of square - Sum of cut - out areas. Given shaded area = 126 \( \text{cm}^2 \). So, \( 225-(77+\frac{5h}{2}) = 126 \).
Simplify: \( 225 - 77-\frac{5h}{2}=126 \) → \( 148-\frac{5h}{2}=126 \).
Subtract 126 from both sides: \( 148 - 126=\frac{5h}{2} \) → \( 22=\frac{5h}{2} \).
Multiply both sides by 2: \( 44 = 5h \).
Divide by 5: \( h=\frac{44}{5}=8.8 \, \text{cm} \).

Answer:

The height \( h \) of the triangle is \( 8.8 \, \text{cm} \).