144. The Right Triangle Blueprint · building with two legs and a perfect corner

Try to break it

Drag B along the horizontal leg and C along the vertical leg. The two legs stretch independently, but ∠A stays at exactly 90° because the legs meet at right angles. Try to tilt that corner — you can't.

How you build it

Construct a right triangle from two legs.

• Mark point A — the right-angle vertex of the triangle.
• Mark point B — the end of the first leg AB.
• Draw segment AB — the first leg of the right triangle.
• Construct a perpendicular ray at A — this will become the second leg.
• Place compass at A. Set radius to the given length of AC. Draw an arc cutting the perpendicular ray.
• Mark point C where the arc meets the perpendicular ray.
• Join B and C to complete the right triangle.

The proof, step by step

Prove that the angle ∠BAC of the constructed triangle is 90°.

• We start by drawing the base AB with the given length.
• At point A, we construct a ray perpendicular to AB.
• On this perpendicular ray, we mark point C such that AC equals the given length.
• Joining B and C completes the triangle ABC.
• Since AC lies on the perpendicular ray from AB, angle BAC is exactly 90 degrees.
• Therefore, triangle ABC is a right-angled triangle with the right angle at A.

Worked example

In a right-angled triangle ABC, right-angled at A, if AB = 6 cm and AC = 8 cm, what is the length of BC?

By Pythagoras theorem, BC² = AB² + AC² = 6² + 8² = 36 + 64 = 100. So BC = 10 cm.

• 10 cm — correct
• 12 cm
• 14 cm
• 16 cm