The triangle inequality

Geometry, grade 8 Lesson 13 of 21Learning

The triangle inequality

Three sticks

Not every three lengths make a triangle.
We have three sticks: 3, 4 and 10 cm long.
We lay down the long stick and fix a short stick to each of its ends – the ends $\displaystyle D$ and $\displaystyle E$ of the short sticks do not meet:

Even lying flat on the long stick, the two short sticks cover only $\displaystyle {3+4}={7}$ cm of the 10. To close a triangle, together they have to be longer than 10 cm.

The shortest path

The segment $\displaystyle AB$ is the shortest path from $\displaystyle A$ to $\displaystyle B$ ; every other path is longer – the path through the vertex $\displaystyle C$ too:

the straight path from $\displaystyle A$ to $\displaystyle B$ is $\displaystyle AB$ long;
the path from $\displaystyle A$ to $\displaystyle B$ through $\displaystyle C$ is $\displaystyle AC+CB$ long;
so $\displaystyle AC+CB>AB$ .

The triangle inequality

The triangle inequality: in every triangle, any two sides add up to more than the third side.

In a triangle with sides $\displaystyle a$ , $\displaystyle b$ and $\displaystyle c$ , three inequalities hold, one for each side:

\displaystyle a+b>c

\displaystyle a+c>b

\displaystyle b+c>a

Checking whether three lengths make a triangle

The longest side alone is already longer than each of the others, so the other two inequalities always hold.

To check whether three lengths make a triangle, the longest side is enough: the triangle exists if it is shorter than the other two together.

5, 7 and 10: $\displaystyle {5+7}={12}$ , and $\displaystyle 12>10$ – there is a triangle.
3, 4 and 10: $\displaystyle {3+4}={7}$ , and $\displaystyle 7<10$ – there is no triangle.
3, 4 and 7: $\displaystyle {3+4}={7}$ – the short sticks lie exactly along the long one, leaving no room for a vertex: there is no triangle.

Exercise 1

Is there a triangle with these sides? Give the reason.

6 cm, 8 cm and 13 cm

2 cm, 5 cm and 8 cm

9 cm, 9 cm and 17 cm

4 cm, 11 cm and 7 cm

The third side: between the difference and the sum

When two sides of a triangle are known, the third side is shorter than their sum and longer than their difference.

\displaystyle a-b<c<a+b

\displaystyle a\ge b

For example, two sides are 9 and 5. By the triangle inequality, $\displaystyle c<9+5$ , that is, $\displaystyle c<14$ .
And $\displaystyle 5+c>9$ : we subtract 5 from both sides and get $\displaystyle c>4$ .
Together: $\displaystyle 4<c<14$ .

Exercise 2

A triangle has two sides: one 6 cm long and one 10 cm long.

Between which numbers is the length of the third side?

The third side is a whole number of centimetres. How many such lengths are possible?

/ – 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

The sides of a triangle are $\displaystyle x$ , $\displaystyle x+3$ and 15 cm long.
The 15 cm side is the longest side.
Write an inequality by the triangle inequality, and solve it.

Exercise 4

Noa’s home is 800 metres from her school in a straight line.
Her home is 300 metres from the grocery in a straight line.
The home, the school and the grocery are not on one straight line.
Between which numbers is the straight-line distance from the grocery to the school?

Formulas: geometry, grade 8

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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
$\displaystyle a\cdot d$ $\displaystyle =$ $\displaystyle b\cdot c$ $\displaystyle \frac{a}{b}=\frac{c}{d}$