Mathematics · Plane geometry · Grade 8

Pythagoras’ theorem

How high up the wall does the ladder reach? Does the rod fit in the box?

The situation

An electrician leans a 5-metre ladder against a wall: its foot is 3 metres from the wall. How high up the wall does the ladder reach? There is no need to measure with a tape — in a right-angled triangle, if two sides are known, the third can be calculated. How, and why exactly so? We count squares, prove it by moving triangles, and then apply it to a distance on a map and to the diagonal of a box.

Topic

Pythagoras’ theorem and its applications. A ladder against a wall, the carpenter’s 3-4-5 rope, a distance on a map and the diagonal of a box

Key ideas

Right-angled triangle
Legs and hypotenuse
Proof

Formulas

a^{2} + b^{2} = c^{2}

Pythagoras’ theorem: in a right-angled triangle the sum of the squares of the legs equals the square of the hypotenuse.

c = \sqrt{a^{2} + b^{2}}, a = \sqrt{c^{2} - b^{2}}

If two sides are known, the third can be found (the ladder: $\sqrt{25 - 9} = 4$ m).

d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}

The distance between two points in the coordinate plane is the hypotenuse of a triangle whose legs are $| x_{2} - x_{1} |$ and $| y_{2} - y_{1} |$ .

D = \sqrt{a^{2} + b^{2} + c^{2}}

The diagonal of a cuboid: the theorem is applied twice (the diagonal of the base, then the height).

Theory

What the parts are called

In a right-angled triangle the sides next to the right angle are the legs ( $a$ , $b$ ), and the longest side opposite it is the hypotenuse ( $c$ ). The theorem: $a^{2} + b^{2} = c^{2}$ . In the first step you see this with squares: $3^{2} = 9$ and $4^{2} = 16$ squares together give $25 = 5^{2}$ squares. Even when the squares cannot be counted (for example $c = \sqrt{13}$ ) the equality holds — the proofs show this.

Proof 1: rearranging

Into a large square of side $a + b$ we place four identical right-angled triangles. In one arrangement a square of side $c$ is left in the middle; moving the triangles, in the second arrangement two squares — $a^{2}$ and $b^{2}$ — are left. The large square and the four triangles are the same in both, so the remaining areas are equal: $c^{2} = a^{2} + b^{2}$ . There is no calculation in this proof — only the conservation of area.

Proof 2: similar triangles

From the right-angle vertex we drop an altitude to the hypotenuse. The two small triangles formed are similar to the large one (they have the same angles). From similarity $\frac{b}{c} = \frac{c_{1}}{b}$ , that is $b^{2} = c \cdot c_{1}$ , and in the same way $a^{2} = c \cdot c_{2}$ . Adding: $a^{2} + b^{2} = c (c_{1} + c_{2}) = c \cdot c = c^{2}$ . Try dragging a vertex of the triangle — the proof works for every right-angled triangle.

The converse theorem and triples

If the sides of a triangle satisfy $a^{2} + b^{2} = c^{2}$ , the angle opposite $c$ is a right angle. That is why a rope with 12 knots (3 + 4 + 5) gives a right angle in building. Such triangles with whole-number sides give Pythagorean triples: 3-4-5 and its multiples (6-8-10 …), but there are others too — find them in the fourth step. All the triples are given by the formula $m^{2} - n^{2}, 2 m n, m^{2} + n^{2}$ ( $m > n$ natural numbers).

Distance and diagonal

In the coordinate plane the distance between the points $A (x_{1}; y_{1})$ and $B (x_{2}; y_{2})$ is the hypotenuse of a triangle whose legs are $| x_{2} - x_{1} |$ and $| y_{2} - y_{1} |$ : $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$ . In space the theorem is applied twice: the diagonal of the base of a box is $d_{1} = \sqrt{a^{2} + b^{2}}$ , then from $d_{1}$ and the height $c$ , $D = \sqrt{d_{1}^{2} + c^{2}} = \sqrt{a^{2} + b^{2} + c^{2}}$ . A 70 cm rod fits inside a 40 × 30 × 50 cm box: $D \approx 70.7$ cm.

About the model

In the drawings the sides are whole numbers or decimals and the calculations are exact: irrational values such as $\sqrt{2}$ are shown rounded (two places after the decimal point). The rod in the box is taken to have zero thickness — a real rod needs a few more millimetres of room.

Did you know?

The Babylonian clay tablet “Plimpton 322” (c. 1800 BC) contains a table of Pythagorean triples — more than a thousand years before Pythagoras.
The theorem has hundreds of proofs: Elisha Loomis’s book “The Pythagorean Proposition” (1940 edition) collects 370 of them; one is by the US President J. Garfield.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.