Question

15 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 = ½(a + b) × 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 \( 15 \, \text{cm} \), so area is \( 15 \times 15 = 225 \, \text{cm}^2 \).

Step2: Calculate area of rectangle

Rectangle: length \( 7 \, \text{cm} \), width \( 4 \, \text{cm} \), area \( 7 \times 4 = 28 \, \text{cm}^2 \).

Step3: Calculate area of trapezium

Trapezium: \( a = 4 \, \text{cm} \), \( b = 6 \, \text{cm} \), \( h = 5 \, \text{cm} \), area \( \frac{1}{2}(4 + 6) \times 5 = 25 \, \text{cm}^2 \).

Step4: Calculate area of parallelogram

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

Step5: Calculate area of triangle (unknown height)

Let triangle area be \( \frac{1}{2} \times 5 \times h \).

Step6: Sum of cut - out areas

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

Step7: Set up equation for shaded area

Shaded area = square area - sum of cut - out areas: \( 225-(28 + 25 + 24+\frac{5h}{2})=126 \).
Simplify: \( 225-(77+\frac{5h}{2}) = 126 \) → \( 225 - 77-\frac{5h}{2}=126 \) → \( 148-\frac{5h}{2}=126 \).
Subtract 126: \( 22=\frac{5h}{2} \).
Multiply by 2: \( 44 = 5h \).
Divide by 5: \( h=\frac{44}{5}=8.8 \, \text{cm} \).

Answer:

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