Using Pythagoras' theorem

Geometry, grade 8 Lesson 5 of 21Learning

Using Pythagoras' theorem

Finding a leg

When the hypotenuse and one leg are known, we find the other leg: we move $\displaystyle b^{2}$ to the other side of Pythagoras’ theorem.

\displaystyle a^2

\displaystyle =

\displaystyle c^2-b^2

Example: the hypotenuse is 13 and one leg is 5. The other leg: $\displaystyle {a}={\sqrt{13^{2}-5^{2}}}={\sqrt{169-25}}={\sqrt{144}}={12}$ .
To find a leg we subtract: the hypotenuse is the longest side, and its square is the largest.

Exercise 1

In each part, the hypotenuse and one leg of a right triangle: hypotenuse 10 and leg 6; hypotenuse 25 and leg 7; hypotenuse 17 and leg 15; hypotenuse 2.5 and leg 1.5.
Find the other leg.

a

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

d

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Example: a ladder against a wall

A ladder 2.5 m long leans against a wall.
Its foot is 0.7 m from the wall.
How high does it reach?

The wall and the floor are perpendicular, so the wall, the floor and the ladder make a right triangle.
The ladder is opposite the right angle, so it is the hypotenuse: 2.5. The legs are the distance from the wall, 0.7, and the height $\displaystyle h$ .
$\displaystyle {h^{2}}={2.5^{2}-0.7^{2}}={6.25-0.49}={5.76}$ .
$\displaystyle {h}={\sqrt{5.76}}={2.4}$ : the ladder reaches a height of 2.4 m.

Exercise 2

A television screen is 48 cm wide and 36 cm high.
The size of a screen is measured along its diagonal.
How long is the diagonal?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A field is a rectangle 80 m long and 60 m wide.
Noa walks from one corner of the field to the opposite corner along the diagonal, instead of along two sides.
How many metres shorter is her way?

The diagonal

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

How much shorter

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A ladder 5 m long leans against a wall and reaches a height of 4.8 m.
How far from the wall is the foot of the ladder?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The converse of Pythagoras’ theorem

This is how builders check a right corner: they mark 3 m along one wall and 4 m along the other. If the two marks are exactly 5 m apart, the corner is right.

The converse: if the sides of a triangle satisfy $\displaystyle {a^{2}+b^{2}}={c^{2}}$ , the triangle is right-angled, and the right angle is opposite the side $\displaystyle c$ .

Why the converse holds

We keep the sides $\displaystyle CA$ and $\displaystyle CB$ the same length, and open and close the angle $\displaystyle C$ between them, like a pair of compasses:

The larger the angle $\displaystyle C$ , the farther apart the ends of the two sides, and the longer the third side.
When the angle is right, the third side $\displaystyle AB$ is the hypotenuse, and its square is exactly $\displaystyle CA^{2}+CB^{2}$ .
At an angle smaller than $\displaystyle 90^{\circ}$ (the dashed line to $\displaystyle D$ ) the third side is shorter, and at an angle larger than $\displaystyle 90^{\circ}$ (to $\displaystyle E$ ) it is longer.
So only at a right angle does the square of the third side equal $\displaystyle CA^{2}+CB^{2}$ .

Checking whether a triangle is right-angled

To check whether a triangle is right-angled, we compare the square of the longest side with the sum of the squares of the other two sides.

Sides 6, 8, 10: $\displaystyle {6^{2}+8^{2}}={36+64}={100}={10^{2}}$ . The triangle is right-angled.
Sides 5, 6, 8: $\displaystyle {5^{2}+6^{2}}={25+36}={61}$ $5^{2} + 6^{2} = 25 + 36 = 61$ , but $\displaystyle {8^{2}}={64}$ $8^{2} = 64$ . The triangle is not right-angled.
- $\displaystyle 64>61$ , so the angle opposite the side 8 is larger than $\displaystyle 90^{\circ}$ : it is an obtuse triangle.

Exercise 5

Is a triangle with these side lengths right-angled? Explain.

a

9, 12, 15

b

7, 8, 11

c

13, 5, 12

d

2, 3, 4

Pythagorean triples

A Pythagorean triple is three whole numbers $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ with $\displaystyle {a^{2}+b^{2}}={c^{2}}$ – the side lengths of a right triangle.

Well-known triples: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25.
Multiplying all three numbers of a Pythagorean triple by the same whole number gives another triple: from 3, 4, 5 we get 6, 8, 10 and also 9, 12, 15.
Why: when each side is 2 times longer, each square is 4 times larger, so the equality between the squares stays.

Exercise 6

Show that 20, 21, 29 is a Pythagorean triple.
Write two Pythagorean triples that come from 5, 12, 13.

Formulas: geometry, grade 8

Angles between parallel lines

Adjacent angles
$\displaystyle \alpha+\beta=180^{\circ}$
Vertical angles are equal
$\displaystyle \alpha=\gamma$
Between parallel lines: corresponding angles are equal, and so are alternate angles
$\displaystyle \alpha=\beta$
Between parallel lines: co-interior angles
$\displaystyle \gamma+\delta=180^{\circ}$

The angles of a triangle

The angles of a triangle
$\displaystyle \alpha+\beta+\gamma=180^{\circ}$
An exterior angle $\displaystyle \delta$ equals the sum of the two interior angles not next to it
$\displaystyle \delta$ $\displaystyle =$ $\displaystyle \alpha+\beta$
In an isosceles triangle the base angles are equal
$\displaystyle \angle B=\angle C$ $\displaystyle AB=AC$
In an equilateral triangle all the angles are equal
$\displaystyle \alpha=\beta=\gamma=60^{\circ}$

Pythagoras' theorem

In a right triangle – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle a^2+b^2$ $\displaystyle =$ $\displaystyle c^2$
Finding a leg – from the hypotenuse and the other leg
$\displaystyle a^2$ $\displaystyle =$ $\displaystyle c^2-b^2$

The sides of a triangle

The triangle inequality – two sides together are longer than the third
$\displaystyle a+b>c$
The third side – between the difference of the other two and their sum
$\displaystyle a-b<c<a+b$ $\displaystyle a\ge b$

Similar triangles

In similar triangles $\displaystyle \triangle ABC\sim\triangle DEF$ the corresponding sides are in one ratio – $\displaystyle k$ the ratio of similarity
$\displaystyle \frac{DE}{AB}=\frac{EF}{BC}=\frac{FD}{CA}=k$
The ratio of the perimeters – the perimeter grows $\displaystyle k$ times too
$\displaystyle P_{DEF}$ $\displaystyle =$ $\displaystyle k\cdot P_{ABC}$
Cross-multiplying a proportion
$\displaystyle a\cdot d$ $\displaystyle =$ $\displaystyle b\cdot c$ $\displaystyle \frac{a}{b}=\frac{c}{d}$