Shapes, collinear points and the area of a triangle

What kind of triangle?

Find the three side lengths with the distance formula. Compare the squares of the lengths, so that there is no rounding.

Example: $A(1, 2)$, $B(5, 2)$, $C(3, 6)$. Then $AB^2 = 16$, $BC^2 = 4 + 16 = 20$ and $CA^2 = 4 + 16 = 20$. Two sides are equal, so the triangle is isosceles. Is it right-angled? $16 + 20 = 36$ is not $20$, so no.

What kind of quadrilateral?

• Parallelogram: the two diagonals have the same midpoint. They cut each other in half.
• Rectangle: a parallelogram with equal diagonals.
• Rhombus: a parallelogram with equal sides.
• Square: equal sides and equal diagonals.

Example. Take $A(1, 2)$, $B(4, 3)$, $C(5, 6)$, $D(2, 5)$.

• The midpoint of $AC$ is $(3, 4)$, and the midpoint of $BD$ is $(3, 4)$. Same midpoint, so $ABCD$ is a parallelogram.
• $AB^2 = 9 + 1 = 10$ and $BC^2 = 1 + 9 = 10$. Adjacent sides are equal, so it is a rhombus.
• $AC^2 = 16 + 16 = 32$ and $BD^2 = 4 + 4 = 8$. The diagonals are not equal, so it is not a square.

The area of a triangle

For a triangle with corners $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$:

Example: $(1, 1)$, $(5, 1)$ and $(3, 6)$. Inside the bars, $1(1 - 6) + 5(6 - 1) + 3(1 - 1) = -5 + 25 + 0 = 20$. Half of $20$ is $10$ square units. As a check, the base is $4$ and the height is $5$, so $\tfrac{1}{2} \times 4 \times 5 = 10$. It agrees.

The bars $| \ |$ mean the absolute value. The area cannot be negative, and the sign inside the bars depends on the order of the corners, so we drop it.

The area of a quadrilateral. Cut it into two triangles by a diagonal, find the area of each, and add.

Move $C$ onto the dashed line, for example to $(4, 2)$. The area drops to $0$: the triangle has flattened into a line.

Collinear points

Three points are collinear if they lie on one straight line. Then they make a triangle with zero area. Another way: the sum of two of the distances equals the third, such as $AB + BC = AC$.

Example: for $A(1, 2)$, $B(3, 6)$ and $C(5, k)$ to be collinear, the area is zero. Inside the bars, $1(6 - k) + 3(k - 2) + 5(2 - 6) = 2k - 20$. This must be $0$, so $k = 10$.

A check: from $A$ to $B$ the line goes $2$ across and $4$ up. From $B$ to $C(5, 10)$ it goes $2$ across and $4$ up. The same step, so it is a straight line. ✓

The centroid

The centroid of a triangle is the point where its three medians meet. Its coordinates are the average of the corners:

Example: for $(0, 0)$, $(6, 0)$ and $(3, 9)$, $G = (3, 3)$. A cardboard triangle balances on a pin at this point.

Practise with new numbers

Try it in Python

This program names a triangle from the squares of its sides.

Common mistakes

• Comparing lengths with rounding. Compare the squares.
• Calling a quadrilateral with equal diagonals a rectangle. It must also be a parallelogram.
• Forgetting the absolute value in the area. The area is not negative.
• Taking the largest square as one of the two legs. In the right-angle test, $c$ is the longest.

Remember these

• Compare squared lengths to name a triangle.
• A parallelogram has diagonals with the same midpoint.
• Collinear points make a triangle of zero area.
• The centroid is the average of the three corners.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A triangle has sides whose squares are $9$, $16$ and $25$. What kind is it?

2. If three points are collinear, what is the area of the triangle they make?

3. Find the area of the triangle with the corners $(0, 0)$, $(4, 0)$ and $(0, 3)$.

4. Are $(0, 0)$, $(1, 2)$ and $(3, 6)$ collinear?