Chapter 4 · Geometry and Equations · Concept 2
4-7 Shapes of Triangles
In radiosurgery, several narrow beams are aimed from different angles to meet at a tumour. If they miss each other and form a small accidental triangle instead of meeting at a point, the treatment fails — so the shape three lines make is not an academic question.
Guiding question
How can equations of straight lines guarantee that several paths meet at one exact point?
Learning outcome. Use coordinates to decide what kind of triangle three points make.
Work out three numbers, then read them
To decide the shape of triangle ABC, calculate (AB)², (BC)² and (CA)². Then:
Where the right angle is.
If (AC)² = (AB)² + (BC)² then the right angle is at B — the corner that AC does not touch. The biggest side is always opposite the right angle, so name the corner the long side runs past, not one of its ends.
Never take the square roots. Every comparison works on the squares, and the squares are usually whole numbers while the sides are surds. Rooting is extra work that makes the arithmetic worse and proves nothing new.
Press the button and watch which tests fire. The red ring goes round the corner that has the right angle — check it against the side lengths each time.
AB²
?
BC²
?
CA²
?
Determine the shape of triangle ABC with A(4, 9), B(0, 6), C(3, 2).
- 1
Work out the first squared side, and do not root it.
(AB)2 = (0 − 4)2 + (6 − 9)2 = 16 + 9 = 25Squared sides are what every test in this lesson compares, and 25 is easier to work with than 5.
Work with a partner. Plot the triangle before calculating anything. A rough sketch tells you which corner LOOKS like the right angle, and the arithmetic then either confirms it or tells you the sketch was misleading.
Question 4 is the odd one out.
No two sides are equal, so it is not isosceles — but 25 + 100 = 125 still holds, so it IS right-angled. Being right-angled and being isosceles are two separate tests, and a triangle can pass one without the other.
A(1, 2), B(2k, k) and C(4, 5). Triangle ABC is isosceles and right-angled at B. Find k.
Naming the wrong corner for the right angle.
If (CA)² = (AB)² + (BC)², the right angle is at B — the corner CA does not touch. The long side is always opposite it.
Taking square roots on the way.
Compare the squares. They are usually whole numbers, and rooting them turns easy arithmetic into surds for no gain.
Stopping at "isosceles".
Once two sides match, still test the sum. Most of these triangles are isosceles AND right-angled, and both words earn marks.
Assuming isosceles and right-angled go together.
Exercise 4 is right-angled with no two sides equal. They are separate tests.
Giving up on a surd coordinate.
In Try 2 the √3 terms cancel in every side. Expand carefully and the answer comes out as a whole number.
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