A centimetre grid is shown. 𝐴 three, five 𝐵 negative five, four, and 𝐶 negative one, negative two are three points. What type of triangle is 𝐴𝐵𝐶? You must show how you reached your answer.
Our first step here is to plot the three points or coordinates on the grid. Remember we go along the corridor and then up the stairs. The first coordinate is the 𝑥-coordinate and the second one is the 𝑦-coordinate. To plot point 𝐴, we go along the 𝑥- axis to three and then up the 𝑦 axis to five. For point 𝐵, we go along to negative five and then up to four. Finally, to plot point 𝐶, we go along to negative one and down to negative two. Joining the three points creates a triangle.
We have been asked to work out what type of triangle 𝐴𝐵𝐶 is. It could be equilateral, isosceles, or scalene. An equilateral triangle has three equal-length sides, an isosceles triangle has two equal-length sides, and a scalene triangle has no equal-length sides. In order to work out which type our triangle is, we need to calculate the length of all three sides of the triangle. We can do this using Pythagoras’s theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the longest side of the triangle or the hypotenuse.
Pythagoras’s theorem only works for right-angled triangles. Let’s firstly consider the length from 𝐵 to 𝐶. The difference between the 𝑥-coordinates of 𝐵 and 𝐶 is four as we go along four squares. The difference between the 𝑦-coordinates four and negative two is six as we go up six squares. This creates a right-angled triangle as shown on the diagram. As 𝐵𝐶 is the longest side or hypotenuse of this right-angled triangle, we can use Pythagoras’s theorem to state that 𝐵𝐶 squared is equal to six squared plus four squared. Six squared is equal to 36 as six multiplied by six is 36. Four squared is equal to 16. Therefore, 𝐵𝐶 squared is equal to 36 plus 16. 36 plus 16 is equal to 52.
We’ve now worked out 𝐵𝐶 squared. However, we want to work out the length of 𝐵𝐶. In order to do this, we need to square root both sides of the equation as square rooting is the opposite or inverse of squaring. Square rooting 𝐵𝐶 squared gives us 𝐵𝐶. This is equal to the square root of 52. As this is a non-calculator paper, we will leave our answer in surd form. We will now repeat this process to work out the length of 𝐴𝐶.
The difference between the 𝑥-coordinates of 𝐴 and 𝐶 three and negative one is four. And the difference between the 𝑦-coordinates of five and negative two is seven. Once again, we have created a right-angled triangle with hypotenuse 𝐴𝐶 such that 𝐴𝐶 squared is equal to seven squared plus four squared. Seven squared is equal to 49 and four squared is equal to 16. 49 plus 16 is equal to 65. Once again, we can square root both sides of this equation. This tells us that the length 𝐴𝐶 in the triangle is equal to the square root of 65 or root 65.
The length of 𝐵𝐶 and the length of 𝐴𝐶 are different. Therefore, we can rule out an equilateral triangle as all three sides cannot now be equal. Finally, we need to calculate the length of 𝐴𝐵. The difference between the 𝑥-coordinates of 𝐴 and 𝐵 is eight as the difference between three and negative five is eight. The difference between the 𝑦-coordinates five and four is one. We have created another right-angled triangle, where 𝐴𝐵 squared is equal to eight squared plus one squared. Eight squared is equal to 64 and one squared is equal to one as one multiplied by one is one. Adding these numbers gives us that 𝐴𝐵 squared is equal to 65. Square rooting both sides of this equation tells us that the length of 𝐴𝐵 is root 65.
We notice here that the length of 𝐴𝐶 and the length of 𝐴𝐵 are equal. As two sides of the triangle are equal in length, we can conclude that it is an isosceles triangle. The triangle 𝐴𝐵𝐶 with coordinates three, five; negative five, four; and negative one, negative two is isosceles.