The Pythagorean theorem: distance from square areas

Where we are on the map

Area formulas now support a theorem about distance.

Begin with a familiar image

Walking 3 blocks east and 4 north does not make a straight distance of 7; the diagonal joins the endpoints directly.

Precise meaning

In a right triangle, a²+b²=c², where c is opposite the right angle. The converse checks a right angle from side lengths.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

right angle → hypotenuse → squares of legs → equation → positive root → estimate

Worked example

c²=3²+4²=25, so c=5; it exceeds either leg and is below their sum.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Using 10 as a leg in c²=6²+10² is wrong when 10 is already the longest side and may be the hypotenuse.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. A 6.5 m ladder stands 2.5 m from a wall. Find its height and check with the 5–12–13 triple.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

A circle gathers all points at one distance from its center and introduces π.

Course contents