The sides of a triangle have lengths 7 cm, 24 cm, 25 cm. Find the area of the triangle in two different ways.

Step-by-step solution

Idea: Heron’s formula works for any triangle. Here 7² + 24² = 625 = 25², so the triangle is also right-angled and its legs give base and height.

1. Heron: s = 7 + 24 + 252 = 28. Area = √(28 × 21 × 4 × 3) = √7056 = 84 cm².1½ marks
2. Right triangle: 7² + 24² = 49 + 576 = 625 = 25², so by the converse of the Baudhāyana–Pythagoras theorem the angle between the 7 cm and 24 cm sides is 90°. Area = 12 × 7 × 24 = 84 cm².1½ marks

Answer to write in the exam

Common mistakes that cost marks

• Using 25 cm as a leg in 12 × base × height; it is the hypotenuse.
• Claiming the triangle is right-angled without checking 7² + 24² = 25².

How this can come in the exam

Try one yourself

Find the area of a triangle with sides 8 cm, 15 cm, 17 cm in two ways.

More questions like this

• If the wheel of a bicycle has a diameter of 60 cm, find how far a cyclist will have travelled after the wheel has rotated 100 times.
• Find the area of a quadrant of a circle whose circumference is 66 cm.
• The wheel of a car has an outer radius of 28 cm. Calculate how far the car travels after one complete turn of the wheel, and how many times the wheel turns during a journey of 1 km.
• Two rectangles have the same area and the same perimeter. Does this mean that they are congruent to each other?
• You know that the area of a parallelogram is base × height. Using this and the figure, show that the area of a trapezium is half the sum of the parallel sides × height, i.e., 12(a + b)h.

All Perimeter and area questions · All maths questions