Grade by Grade
E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

Triangle Properties and the Centres of a Triangle

E1.2 A1About 70 minutes
Learning goals
  • I can use the angle sum, the exterior angle property and the properties of isosceles and equilateral triangles to find unknown angles.
  • I can use the triangle inequality to decide whether three lengths can make a triangle.
  • I can locate the circumcentre, incentre and centroid of a triangle and describe what each one does.
  • I can draw the circumscribed and inscribed circles of a triangle.

Words to know

exterior angle
The angle between one side of a triangle and the extension of the next side.
isosceles triangle
A triangle with at least two equal sides; the angles opposite them (base angles) are equal.
triangle inequality
In any triangle, the sum of any two sides is greater than the third side.
circumcentre
Where the perpendicular bisectors of the sides meet. It is the centre of the circle through all three vertices (the circumscribed circle).
incentre
Where the angle bisectors meet. It is the centre of the circle that touches all three sides (the inscribed circle).
median
A segment from a vertex to the midpoint of the opposite side.
centroid
Where the three medians meet: the balance point of a triangle.

A hook: the fair location

Three towns want to build one shared arena the same distance from all three. Another group wants to place the biggest possible round fountain inside a triangular park. A third wants to hang a triangular sign so it balances perfectly.

Each problem has a special point in the triangle. By the end of this lesson you will know all three.

Remember from Grade 8

The interior angles of a triangle add to 180°, and the interior angles of a polygon with n sides add to (n − 2) × 180°.

New properties

  • Exterior angle property: an exterior angle equals the sum of the two interior angles that are not next to it. (Reason: the exterior angle and the interior angle at that vertex make 180°, and so do all three interior angles.)
  • Isosceles triangle: the two base angles are equal. Equilateral triangle: all angles are 60°.
  • Sides and angles: the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.
47°68°65°115°ABCexterior angle at B= 47° + 68° = 115°(the two interior anglesthat are not at B)
The exterior angle at B (115°) equals the sum of the interior angles at A and C (47° + 68°). It is also 180° − 65°.

The triangle inequality

Three sticks make a triangle only if any two together are longer than the third. For lengths a ≤ b ≤ c, you only need to check a + b > c.

If you know two sides, the third side x must be between their difference and their sum. For sides 5 and 9: 9 − 5 < x < 9 + 5, so 4 < x < 14.

5433 + 4 > 5: it closes7232 + 3 < 7: the ends can't meet
Left: sides 3, 4 and 5 meet to make a triangle. Right: sides 2 and 3 are too short to reach each other across 7, so no triangle is possible (the arcs show every place their ends could swing).

Three centres of a triangle

  • Circumcentre: draw the perpendicular bisector of each side. They meet at one point that is the same distance from all three vertices. A circle centred there passes through all three vertices: the circumscribed circle. (Arena problem!) In a right triangle, the circumcentre is the midpoint of the hypotenuse, because the hypotenuse is a diameter (angle in a semicircle).
  • Incentre: draw the angle bisectors. They meet at a point the same distance from all three sides. A circle centred there touches each side once: the inscribed circle. (Fountain problem!)
  • Centroid: draw the three medians. They meet at the balance point, two-thirds of the way along each median from its vertex. (Sign problem!)

In an equilateral triangle, all three centres are the same point.

dot = circumcentreperpendicular bisectorsdot = incentreangle bisectors
Left: the perpendicular bisectors of the sides meet at the circumcentre, the centre of the circle through all three vertices. Right: the angle bisectors meet at the incentre, the centre of the circle that touches all three sides.
Worked example

Example 1: Exterior angle

Two interior angles of a triangle are 47° and 68°. Find the exterior angle at the third vertex and the third interior angle.

  1. Exterior angle = 47° + 68° = 115°.
  2. Third interior angle = 180° − 115° = 65°.
  3. Check: 47 + 68 + 65 = 180. ✓

Exterior 115°, interior 65°

Worked example

Example 2: Isosceles triangle

An isosceles triangle has a vertex angle of 40°. Find the base angles. Then: another isosceles triangle has base angles of 52°. Find its vertex angle.

  1. Base angles are equal: (180° − 40°) ÷ 2 = 70° each.
  2. Second triangle: vertex = 180° − 2 × 52° = 76°.

70° and 70°; 76°

Worked example

Example 3: Can it be a triangle?

Can sides 5 cm, 7 cm and 11 cm make a triangle? What about 4 cm, 7 cm and 12 cm?

  1. Check the two shortest sides against the longest.
  2. 5 + 7 = 12 > 11 ✓, so yes.
  3. 4 + 7 = 11, which is less than 12 ✗, so no.

Yes for 5, 7, 11; no for 4, 7, 12.

Worked example

Example 4: The fair arena

Three towns form a right triangle: the roads between them are 6 km, 8 km and 10 km. Where should the arena go, and how far is it from each town?

  1. 6² + 8² = 100 = 10², so it is a right triangle with hypotenuse 10 km.
  2. The circumcentre of a right triangle is the midpoint of the hypotenuse.
  3. Distance to each town = the circumradius = 10 ÷ 2 = 5 km.

At the midpoint of the 10 km road, 5 km from each town.

Worked example

Example 5: An equilateral triangle

An equilateral triangle has sides of 6 cm. Find its height, its circumradius and its inradius.

  1. The height splits the base into 3 cm and 3 cm. h² + 3² = 6², so h² = 27 and h ≈ 5.20 cm.
  2. All centres are the same point, found two-thirds of the way down each median (here the median is the height).
  3. Circumradius R = ⅔ × 5.20 ≈ 3.46 cm. Inradius r = ⅓ × 5.20 ≈ 1.73 cm.

h ≈ 5.20 cm, R ≈ 3.46 cm, r ≈ 1.73 cm (R is exactly 2r).

Watch out!

  • Circumcentre uses perpendicular bisectors of sides; incentre uses angle bisectors. Don't mix them up.
  • The circumcentre can be outside the triangle (for an obtuse triangle). The incentre and centroid are always inside.
  • For the triangle inequality, "equal" is not enough: 4 + 7 = 11 means the sides lie flat, not a triangle.
Hands-on activity

Balance and fold

You need: a cereal box panel or thin cardboard, a ruler, a pencil, scissors (ask a grown-up to help with cutting), and paper.

  1. Draw any large triangle on the cardboard and cut it out.
  2. Find the midpoint of each side. Draw the three medians. They should meet at one point: the centroid.
  3. Balance the triangle on the eraser end of a pencil at the centroid. It should balance!
  4. On paper, draw another triangle. Fold the paper so two vertices land on each other; the fold is a perpendicular bisector. Do this for all three sides. The folds meet at the circumcentre.
  5. Measure from the circumcentre to each vertex. The distances should be equal.

What to look for: the medians and the folds each meet at one point (within a millimetre or two).

Just for fun

Why was the obtuse triangle always upset?

Because it was never right.

Your turn

1

Two angles of a triangle are 38° and 74°. Find the third.

2

An exterior angle is 130°. One of the two interior angles not next to it is 55°. Find the other.

3

An isosceles triangle has a vertex angle of 50°. Find each base angle.

4

Can sides 3 cm, 4 cm and 8 cm make a triangle?

5

Two sides of a triangle are 6 cm and 10 cm. What lengths are possible for the third side?

6

A triangle has angles 30°, 60° and 90°. Which side is longest?

7

Which centre is the same distance from all three vertices?

More practice: warm-up

1

What is each angle of an equilateral triangle?

2

What do the interior angles of a triangle add to?

3

In a right triangle, what do the two acute angles add to?

4

What is the point where the medians meet called?

5

What is the centre of the inscribed circle called?

More practice: on your own

1

An isosceles triangle has base angles of 52°. Find its vertex angle.

2

A right triangle has legs 5 cm and 12 cm. Find the radius of its circumscribed circle.

3

A triangle has angles 50°, 60° and 70°. Find its three exterior angles (one at each vertex) and their sum.

4

Three cabins form a triangle. Explain how to find the spot for a shared well the same distance from all three.

5

An equilateral triangle has sides of 12 cm. Find its height and the radius of its circumscribed circle, to two decimals.

Challenge

1

In an isosceles triangle, the exterior angle at one base vertex is 110°. Find all three interior angles.

2

A triangle has sides x, x + 2 and 10 (in cm). For which values of x is it a real triangle?

3

Use a circle property to explain why the circumcentre of a right triangle is the midpoint of the hypotenuse.

Check your understanding

Quiz

1.

Two interior angles of a triangle are 50° and 75°. What is the exterior angle at the third vertex?

2.

Which set of lengths can form a triangle?

3.

The circumcentre is where which lines meet?

4.

An isosceles triangle has a vertex angle of 100°. Each base angle is:

5.

A right triangle has a hypotenuse of 14 cm. What is the radius of its circumscribed circle?

6.

Which point is the balance point of a triangle?

For parents and teachers

Big idea: E1.2 includes triangle properties. Students extend Grade 8 angle work with the exterior angle property, isosceles triangles, the triangle inequality and the three triangle centres, which are used to draw inscribed and circumscribed circles in designs.

Common difficulties:

  • Mixing up circumcentre and incentre. Memory trick: circumference goes through the vertices; the incircle stays inside, touching the sides.
  • Triangle inequality with equal sums. 4 + 7 = 11 does not make a triangle with 11.
  • Rounding too early in the equilateral triangle problems.

How to help: Do the balancing activity together and supervise the cutting. Ask your teen to explain why the folds meet at one point.