The idea, step by step
Connect the three side lengths
For a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. Label the hypotenuse before substituting, because it occupies a different role in the equation.
The theorem concerns lengths, so choose the positive square root. Keep an exact root until the final rounding when the root is not a whole number.
Subtract when a leg is missing
If the hypotenuse is known, subtract the square of the known leg from its square. Adding would produce a missing leg longer than the hypotenuse.
Use the triangle’s context to check an answer: a ladder length is the hypotenuse, while a vertical wall and level ground form the legs. In a rectangle, a diagonal and two sides make a right triangle.
A common trap
Using a²+b²=c² without checking which length is c.
Instead: Place the side opposite the right angle in the hypotenuse position, then rearrange before calculating.
See the thinking
Worked example 1
A right triangle has legs 9 m and 12 m. Find the hypotenuse.
- Let c be the hypotenuse: c²=9²+12².
- c²=81+144=225.
- c=√225=15 m. It exceeds both legs.
Worked example 2
A 10 m ladder rests on level ground against a vertical wall. Its foot is 6 m from the wall. How high does it reach?
- The 10 m ladder is the hypotenuse.
- h²=10²−6²=100−36=64.
- h=8 m. Adding the squares here would be incorrect.
Check your understanding
A right triangle has hypotenuse 17 cm and one leg 8 cm. Find the other leg.
Show answer
15 cm
√(17²−8²)=√225=15 cm.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- Find the hypotenuse of a right triangle with legs 5 cm and 12 cm.2 marks
- A right triangle has hypotenuse 13 m and one leg 5 m. Find the other leg.2 marks
- Find the diagonal of a square with side 6 cm. Give an exact answer and then one decimal place.3 marks
- A rectangular field is 9 m by 12 m. How far is it between opposite corners?2 marks
- A 15 m cable reaches from the top of a vertical post to the ground 9 m away. Find the post’s height.2 marks
- A student finds a missing leg in a triangle with hypotenuse 10 and other leg 6 by calculating √(100+36). Explain and correct the error.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Find the hypotenuse of a right triangle with legs 5 cm and 12 cm.2 marks
c²=25+144=169.
c=13 cm.
13 cm - A right triangle has hypotenuse 13 m and one leg 5 m. Find the other leg.2 marks
b²=169−25=144.
b=12 m.
12 m - Find the diagonal of a square with side 6 cm. Give an exact answer and then one decimal place.3 marks
d²=6²+6²=72.
d=√72=6√2≈8.5 cm.
6√2 cm; 8.5 cm - A rectangular field is 9 m by 12 m. How far is it between opposite corners?2 marks
The diagonal is the hypotenuse.
d=√(9²+12²)=15 m.
15 m - A 15 m cable reaches from the top of a vertical post to the ground 9 m away. Find the post’s height.2 marks
h²=15²−9²=144.
h=12 m.
12 m - A student finds a missing leg in a triangle with hypotenuse 10 and other leg 6 by calculating √(100+36). Explain and correct the error.3 marks
The unknown leg cannot exceed the hypotenuse.
b²=100−36=64, so b=8.
Subtract instead: the missing leg is 8 units.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: practical trigonometry, including similarity, right triangles, triangle area, sine/cosine rules and contextual interpretation.
Official curriculum source ↗