Constructions
145. The Hypotenuse Rule · Constructing right triangles from side & hypotenuse
The angle at A is always 90°, making BC the hypotenuse.
To construct a right triangle given hypotenuse and one side: use the fact that an angle inscribed in a semicircle is 90°. Draw the hypotenuse BC as a diameter; A on the semicircle gives ∠A = 90°. Combined with the side length, this fixes the triangle.
What this lesson covers
Try to break it
Drag C up or down the vertical line. BC always stretches to be the longest side — it never drops below AB. Try to make BC shorter than AB; you'd have to break the 90° at A first. The hypotenuse is always opposite the right angle, and always the longest side.
How you build it
Construct a right triangle from side and hypotenuse.
- Mark point A. The right angle of the triangle will be at A.
- Mark point B to the right of A. AB is the given side of the right triangle.
- Draw the line segment from A to B.
- At point A, construct a perpendicular ray. The arc from B will mark C on this ray.
- With B as centre and the hypotenuse length as radius, draw an arc that cuts the perpendicular at A.
- Mark point C where the arc crosses the perpendicular. AC is the other leg of the triangle.
- Join B and C to complete the right triangle. BC is the hypotenuse.
The proof, step by step
Prove that the constructed triangle is right-angled with BC as its hypotenuse.
- We are given AB and hypotenuse BC, with ∠A = 90°.
- We construct a perpendicular at A, ensuring ∠CAB = 90°.
- We swing an arc from B with radius BC to locate C on the perpendicular.
- Joining BC completes ΔABC. Since ∠A = 90°, BC is the hypotenuse by definition.
Worked example
In a right-angled triangle ABC, ∠A = 90°. If AB = 6 cm and BC = 10 cm, what is the length of AC?
Using Pythagoras theorem: BC² = AB² + AC². So, 10² = 6² + AC² → 100 = 36 + AC² → AC² = 64 → AC = 8 cm.
- 4 cm
- 6 cm
- 8 cm — correct
- 16 cm