A triangle has vertices at the points 𝐴: four, two; 𝐵: six, two; and 𝐶: five, negative one. Firstly, work out the lengths of the sides of the triangle. Give your answers as surds in their simplest form. Secondly, what type of triangle is 𝐴𝐵𝐶?
So we’ve been given the coordinates of the three vertices of a triangle. And our first task is to calculate the lengths of the sides. In order to do this, we need to recall the distance formula for calculating the distance between two points on a coordinate grid.
This tells us the distance between the two points with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two can be found by taking the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared.
This is in fact an application of the Pythagorean theorem to calculate the hypotenuse of a right-angled triangle, in which the horizontal length is 𝑥 two minus 𝑥 one and the vertical length is 𝑦 two minus 𝑦 one.
Let’s now apply this to calculating the lengths of the three sides of this triangle. Let’s begin with the side 𝐴𝐵. Now an actual fact, this side is horizontal as both points of the same 𝑦-coordinate. Therefore, its length is just the difference between its 𝑥-coordinates. The difference between six and four is two. And so the length of 𝐴𝐵 is two.
You could of course apply the distance formula. And it would give the same result. But it’s unnecessarily complicated. Next, let’s calculate the length of the side 𝐴𝐶. And we will need the distance formula here.
Substituting the coordinates of 𝐴 and 𝐶 into the distance formula, we have that the length of 𝐴𝐶 is equal to the square root of five minus four squared plus negative one minus two squared. This gives the square root of one squared plus negative three squared.
One squared is one. And negative three squared is nine. So we have the square root of one plus nine, which is the square root of 10. Now this can’t be simplified any further as 10 doesn’t have any square factors. So this surd is in its simplest form.
Now we consider the third side, 𝐵𝐶. Substituting the coordinates for these points gives the square root of five minus six squared plus negative one minus two squared. This is equal to the square root of negative one squared plus negative three squared. Negative one squared is one. And negative three squared is nine. So we have the square root of one plus nine, which again simplifies to the square root of 10.
So we’ve calculated the lengths of the three sides of the triangle. 𝐴𝐵 is two units. 𝐴𝐶 is root 10 units. And 𝐵𝐶 is also root 10 units. Now let’s consider the second part: What type of triangle is 𝐴𝐵𝐶? Well, you will have noticed in the working out for the previous part that two of the sides of this triangle are the same length. 𝐴𝐶 is equal to 𝐵𝐶.
However, the third side of the triangle, 𝐴𝐵, is different. This means that our triangle has two equal sides. And therefore, it must be an isosceles triangle. So I’ll answer then to the two parts of the problem. 𝐴𝐵 is equal to two. 𝐴𝐶 is equal to root 10. 𝐵𝐶 is also equal to root 10. And 𝐴𝐵𝐶 is an isosceles triangle.