12.04 Triangles

Whenever three points do not lie on the same line, we can connect them together with three segments. This three-sided shape is called a triangle. Three angles are formed at the same time (which is how the shape gets its name).

The kinds of angles that are formed lets us classify different types of triangles:

• If all the angles are acute, the triangle is an acute triangle.

• If one of the angles is a right angle, the triangle is a right-angled triangle.

• If one of the angles is obtuse, the triangle is an obtuse triangle.

The lengths of the sides allow us to classify different types of triangles in a completely different way:

• If all the sides have different lengths, the triangle is a scalene triangle

• If at least two sides have the same length, the triangle is an isosceles triangle

• A special kind of isosceles triangle is the equilateral triangle, where all three sides have the same length.

Examples

Example 1

Consider the triangle below.

a

Which of the following words describes this triangle?

A

Scalene

B

Equilateral

C

Isosceles

Worked Solution

Create a strategy

Take note of the sides marked as equal.

Apply the idea

The answer is option C: isosceles, because two sides of the triangle were marked equal.

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b

Which of the following words also describes this triangle?

A

Acute

B

Right-angled

C

Obtuse

Worked Solution

Create a strategy

Look at the size of the angles.

Apply the idea

All the angles of the triangle are acute angles. So the correct answer is A: acute.

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Example 2

Select all isosceles triangles:

A

B

C

D

E

F

Worked Solution

Create a strategy

Count how many equal sides there are.

Apply the idea

Among the choices, only options A, C, and D are the isosceles triangles as they have two (or more) sides of the same length.

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In any triangle the longest side will always be opposite the largest angle. Same with the smallest side and the smallest angle.

If we increase one side while keeping the other two sides the same size the side that is getting longer will also have the opposite angle get bigger. A triangle that has two sides that are the same length means the opposite angles must also be equal.

Consider these side lengths. Can these sides be arranged into a triangle?

These three side lengths can not be made into a triangle with these side lengths no matter how we arrange them.

If we replace one of the sides with a longer side it is now possible to make these three sides into a triangle.

We can compare the two sets of sides to see what is the defining difference.

We can see that for any side we look at, the other two side lengths when combined are longer.

For the impossible sides, two of the sides we can choose the other two sides combined are longer. However, looking at the longest side, the two shorter sides combined are still smaller than the longest side. This is the condition which determines whether a triangle is possible or not. For a triangle the combined length of each pair of sides is longer than the remaining side.

Examples

Example 3

Is it possible to form a triangle with side lengths 13,\,7, and 6?

Worked Solution

Create a strategy

Create a table to compare the side lengths.

Apply the idea

Side	Sum of remaining sides	Compare
6	7+13=20	7+13>6
7	6+13=19	6+13>7
13	6+7=13	6+7=13

In the final row, the side of 13 is equal to the sum of the other two sides, so the triangle is impossible.

The three sides of length 13,\,7, and 6 cannot make a triangle.

Reflect and check

For scalene triangles we only have to check that the smallest two sides are bigger than the largest side, in this case 6+7=13 so the triangle is impossible.

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Example 4

Which side is the shortest side of the triangle?

Worked Solution

Create a strategy

Use the fact that the shortest side is opposite of the smallest angle.

Apply the idea

Based from the given diagram, the smallest angle is \angle A which means that the shortest side is opposite that angle which is BC.

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