8 min readConcept sheetPart III proof or word problem (5 marks); Part I and Part II

Pythagoras theorem and its converse

In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides. The converse tests for a right angle.

Pythagoras theoremIn ΔABC with ∠A = 90°, BC² = AB² + AC²

ConverseIf BC² = AB² + AC², then ∠A = 90°

HypotenuseThe side opposite the right angle; always the longest side

Pythagorean triples3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25

How to draw the figure and prove the theorem

Figure: draw ΔABC with the right angle at A. Draw AD perpendicular to BC. Label A, B, C, D and mark the right angles at A and D.

1
Given: in ΔABC, ∠BAC = 90°. To prove: BC² = AB² + AC². Construction: draw AD ⊥ BC.
2
In ΔABC and ΔDBA: ∠B is common and ∠BAC = ∠BDA = 90°. So ΔABC ~ ΔDBA.
AA criterion
3
Hence AB/BD = BC/AB, so AB² = BC × BD ... (1)
4
In ΔABC and ΔDAC: ∠C is common and ∠BAC = ∠ADC = 90°. So ΔABC ~ ΔDAC.
AA criterion
5
Hence BC/AC = AC/DC, so AC² = BC × DC ... (2)
6
Add (1) and (2): AB² + AC² = BC × BD + BC × DC = BC × (BD + DC).
7
BD + DC = BC, so AB² + AC² = BC × BC = BC². Hence proved.

Theorem vs converse

Pythagoras theoremStart with a right angle. Conclude: (hypotenuse)² = (one side)² + (other side)².

ConverseStart with the lengths. If (longest side)² = sum of the squares of the other two, the angle opposite the longest side is 90°.

To test a triangle, always square the longest side and compare it with the sum of the other two squares.

Example 1: is it a right triangle?

Problem

Check whether each triangle is right-angled. First triangle: 9 cm, 40 cm, 41 cm. Second triangle: 8 cm, 15 cm, 18 cm.

Working

First triangle: longest side 41, and 41² = 1681.
9² + 40² = 81 + 1600 = 1681. They are equal, so it is right-angled (converse of Pythagoras).
Second triangle: longest side 18, and 18² = 324.
8² + 15² = 64 + 225 = 289. Since 289 ≠ 324, it is not right-angled.

Answer

9, 40, 41: right-angled. 8, 15, 18: not right-angled.

Example 2: a ladder against a wall (word problem)

Problem

A 6.5 m ladder leans against the wall of a house in Madurai. Its foot is 2.5 m from the wall. How high up the wall does it reach?

Working

The wall, the ground and the ladder form a right triangle. The ladder is the hypotenuse.
h² + 2.5² = 6.5²
h² = 42.25 − 6.25 = 36
h = √36 = 6

Answer

The ladder reaches 6 m up the wall.

Example 3: two poles (word problem)

Problem

Two poles of heights 16 m and 9 m stand on level ground, 24 m apart. Find the distance between their tops.

Working

Draw a horizontal line from the top of the short pole to the tall pole. This makes a right triangle.
Vertical side = 16 − 9 = 7 m. Horizontal side = 24 m.
d² = 24² + 7² = 576 + 49 = 625
d = √625 = 25

Answer

The tops are 25 m apart.

In real life

Masons check a right-angled corner with a 3-4-5 rope: 3 m on one wall, 4 m on the other, 5 m across. TV and phone screens are sold by their diagonal length.

Exam angle

1 marks: “Which set of sides makes a right triangle: 6, 8, 11 or 7, 24, 25?” 7² + 24² = 49 + 576 = 625 = 25². So 7, 24, 25 is a right triangle.

5 marks: “State and prove Pythagoras theorem.” Statement and figure with AD ⊥ BC. ΔABC ~ ΔDBA gives AB² = BC × BD. ΔABC ~ ΔDAC gives AC² = BC × DC. Add: AB² + AC² = BC².

2 marks: “Ravi cycles 8 km east and then 6 km north from his village. How far is he from the village now?” The paths form a right triangle with sides 8 km and 6 km. d² = 64 + 36 = 100, so d = 10 km.

Remember it

Longest side squared = the other two squared and added. Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.