Geometry

Pythagoras

Follow the areas on a right triangle.

Predict, test and explain

Step 1 of 3

Predict

A right triangle has short sides 3 and 4. What is the hypotenuse?

9 + 16 = 25

Scroll the diagram sideways to see every label.

a^2+b^2=c^2

c = 5

Explore and compare

Can you break the rule?

Doubling both short sides doubles the hypotenuse.

Always, sometimes or never? Test it in the simulation.

Hint

Save the starting triangle, then double each short side.

Check the reasoning

Always. Both square areas become four times as large. Their sum does too, so its square root doubles.

$\sqrt{(2a)^2+(2b)^2}=2\sqrt{a^2+b^2}$

One counterexample disproves “always”. Examples alone do not prove it.

Make a right-angled triangle with this hypotenuse.

$c=10$

Predict

Try it in the simulation. Change one thing at a time.

Hint

Try doubling each short side of the starting triangle.

Explain

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

A right triangle has perpendicular sides of 5 cm and 12 cm. Find its hypotenuse.

Hint

Square both perpendicular lengths, add them, then take the positive square root.

Reveal answer

The hypotenuse is 13 cm.

$c=\sqrt{5^2+12^2}=\sqrt{169}=13\,\text{cm}$

Hint 1

Identify the right angle before choosing the hypotenuse.

Hint 2

Compare the square areas on the three sides.

Hint 3

For a right triangle, the hypotenuse square equals the sum of the other two squares.

Worked example

This example uses fixed values, separate from the diagram controls.

A right triangle has perpendicular sides of 3 cm and 4 cm. Find its hypotenuse.

Answer: 5 cm.

Connect this idea

Watch out: Add the squared lengths, not the lengths; the hypotenuse faces the right angle.