266. Half the Rectangle · Area of a right‑angled triangle

Try to break it

Drag A along the vertical leg to change a, and C along the horizontal leg to change b. The right triangle is exactly half the rectangle with sides a and b, so its area is always ½·a·b. Pull a or b toward 0 and the triangle collapses — the area smoothly drops to 0.

How you build it

Construct a right-angled triangle from its two legs and see it is half a rectangle.

• Point tool: mark B — the right-angle corner.
• Point tool: mark C — the far end of the base.
• Segment tool: join B to C — this is leg b.
• Perp tool: click B (on BC), then click upward — leg BA rises perpendicular to BC.
• Arc tool: centre B, swing an arc that crosses the perpendicular — its radius is leg a.
• Point tool: mark A where the arc meets the perpendicular leg — BA is now leg a.
• Segment tool: join A to C — the hypotenuse. Triangle ABC is right-angled at B, with area ½ × a × b.

The proof, step by step

Prove that the area of a right triangle is half the product of its legs.

• Consider a rectangle with sides equal to the legs a and b.
• The area of this rectangle is a × b.
• The right‑angled triangle ABC is exactly half of this rectangle.
• Therefore, Area of ΔABC = ½ × a × b.

Worked example

In a right‑angled triangle, the sides containing the right angle measure 9 cm and 12 cm. Find the area of the triangle.

Area = ½ × base × height = ½ × 9 × 12 = 54 cm².

• 54 cm² — correct
• 108 cm²
• 72 cm²
• 36 cm²