Applications of Pythagoras' theorem
Apply Pythagoras' theorem to solve real-world problems involving distances, areas and composite shapes.
Worked examples
Height reached by a ladder
Straightforward
Problem
A ladder m long leans against a vertical wall. The foot of the ladder is m from the base of the wall. How high up the wall does the ladder reach?
1
Draw a right-angled triangle. The ladder is the hypotenuse ( m), the ground distance is one leg ( m), and the height reached is the unknown leg ().
m, m, find
2
Rearrange Pythagoras' theorem to find the unknown leg.
3
Take the square root.
m
Answer
The ladder reaches m up the wall.
Wire connecting tops of two poles
Moderate
Problem
Two telephone poles are m apart. One pole is m tall and the other is m tall. A wire connects their tops. Find the length of the wire.
1
Identify the right-angled triangle. The horizontal distance between poles is one leg. The vertical difference between the pole tops is the other leg.
Horizontal leg m. Vertical leg m.
2
Apply Pythagoras' theorem to find the wire length (the hypotenuse).
3
Take the square root.
m
Answer
The wire is m long.
Area of an equilateral triangle
Challenging
Problem
An equilateral triangle has sides of length cm. Find the area of the triangle, to 2 decimal places.
1
Drop a perpendicular from the top vertex to the base. This creates two congruent right-angled triangles. The hypotenuse of each is cm and one leg is half the base, which is cm.
Hypotenuse cm, base leg cm, find height
2
Use Pythagoras' theorem to find the height.
, so cm
3
Use the area formula for a triangle with the full base and the height found.
cm
Answer
cm
Practise
Q1·Straightforward
A ladder m long leans against a vertical wall. The foot of the ladder is m from the base of the wall. How high up the wall does the ladder reach?
Explanation
Using Pythagoras' theorem: , so m.
Q2·Straightforward
A rectangular paddock is m wide and m long. Find the length of the diagonal fence line, in metres.
Explanation
, so m.
Q3·Straightforward
A guy wire is attached to the top of a m vertical pole and anchored to the ground m from the base of the pole. Find the length of the wire, in metres.
Explanation
, so m.
Q4·Straightforward
A square garden has a side length of m. Find the length of its diagonal, to 2 decimal places.
Explanation
, so m.
Q5·Moderate
A boat leaves a marina and travels km due east, then km due north. Find the straight-line distance from the marina to the boat's final position, in kilometres.
Explanation
, so km.
Q6·Moderate
A ramp rises m vertically over a horizontal distance of m. Find the length of the ramp surface, to 2 decimal places.
Explanation
, so m.
Q7·Moderate
A rectangular TV screen is cm wide. The diagonal of the screen measures cm. Find the height of the screen, in centimetres.
Explanation
, so cm.
Q8·Moderate
Two telephone poles are m apart. One pole is m tall and the other is m tall. A wire connects the tops of the two poles. Find the length of the wire, in metres.
Explanation
The vertical difference between the pole tops is m. The horizontal distance is m. So , giving m.
Q9·Challenging
An equilateral triangle has sides of length cm. Find the area of the triangle, to 2 decimal places.
Explanation
Dropping a perpendicular halves the base, giving a right-angled triangle with hypotenuse cm and base cm. Height: , so cm. Area cm.
Q10·Challenging
A rectangular box is cm long, cm wide and cm tall. Find the length of the space diagonal (the straight-line distance from one corner to the opposite corner), to 2 decimal places.
Explanation
Step 1 — diagonal of the base: . Step 2 — space diagonal: , so cm.
Q11·Challenging
An isosceles triangle has two equal sides of m and a base of m. Find the area of the triangle, in square metres.
Explanation
The perpendicular bisects the base, creating a right-angled triangle with hypotenuse m and one leg m. Height: , so m. Area m.
Q12·Challenging
A ship leaves port and travels km south, then km east, then km north. Find the straight-line distance from the port to the ship's final position, to 2 decimal places.
Explanation
Net southward displacement: km. Eastward displacement: km. Straight-line distance: , so km.