Chapter 4 · Geometry and Equations · Concept 2
4-13 Area of a Triangle
Half base times height — the formula you have had since primary school. The difference here is that nobody gives you the base or the height, so you have to build both out of the last five lessons.
Guiding question
How much surface is inside three points?
Learning outcome. Find the area of a triangle from the coordinates of its corners.
Old formula, two new ingredients
area = 12 × base × height
the base
the distance between two of the points
lesson 4-3
the height
the distance from the third point to the LINE through the other two
lesson 4-8, then lesson 4-12
So the order is fixed, and it is three steps.
Find AB. Find the equation of the line AB. Find the distance from C to that line. The middle step produces nothing you were asked for — it exists only so the third one is possible, and skipping it is what makes this lesson feel impossible.
Expect the roots to cancel.
The base is usually something like 2√10 and the height something like √10 or 7/√5. Their product loses the root, so the area is nearly always a whole number. If yours is not, check the base and the height before writing it down.
Press through the three stages. The dashed blue line never appears in the answer — but without it the green height could not be measured at all.
Find the area of triangle ABC with A(1, 1), B(3, 7), C(−3, −1).
- 1
Choose a base. AB is as good as any — pick whichever pair looks tidiest.
AB = √((3 − 1)2 + (7 − 1)2) = √40 = 2√10Any of the three sides can be the base, as long as the height is measured to the point left over.
Work with a partner. Choose different bases — one of you uses AB, the other BC. You must get the same area, and comparing the two routes shows that the choice of base really does not matter.
Part (b) looks frightening and is not.
The base is √82 and the height is 68/√82 — neither tidies on its own. But their product is 68, so the area is 34. Do not tidy the two pieces separately; multiply them and watch the √82 cancel.
In reconstructive burn surgery, an imaging system maps a wound onto a grid where 1 unit = 1 mm. The wound has corners at A(−2, 1), B(1, 5) and C(−21, −21). Find its area, so the right amount of donor tissue can be taken.
Using the distance from C to A or B as the height.
The height is measured to the LINE through A and B, at a right angle — not to either corner.
Skipping the line equation.
It never appears in the answer, so it feels like wasted work. Without it there is nothing for the height to be measured against.
Tidying the base and height separately.
√82 and 68/√82 look horrible apart and cancel beautifully together. Multiply first.
Forgetting the half.
base × height is a rectangle. The triangle is half of it.
Picking an awkward base.
Any side works, so choose the pair with the tidiest numbers. In the Challenge, AB gives 5 and the other pairs give surds.
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