What is the kind of triangle that the points 𝐴 negative two, negative seven; 𝐵 one, six; and 𝐶 nine, six form with respect to its sides?
Let’s begin by plotting these three coordinates on a grid. In order to find the type of triangle according to the sides, we need to establish if we have two sides equal, three sides equal, or no sides equal. This means we’ll need to work out the length of each side and compare.
Starting with the simplest horizontal length between 𝐵 and 𝐶, we know that since it goes from the coordinate 𝐵 with 𝑥-value one to the coordinate 𝐶 with 𝑥-value nine. Then the length must be eight units. It might be clear at this point that we’re unlikely to have any other length of eight and that the other two lengths probably aren’t equal. But let’s see if we can prove it mathematically.
We need to find the length of each side in the triangle. And we can do this using the distance formula. This tells us that the distance between two coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two is equal to the square root of 𝑥 two minus 𝑥 one all squared plus 𝑦 two minus 𝑦 one all squared. Starting with the length of line 𝐴𝐵, we have 𝑥 one equals negative two. 𝑦 one equals negative seven. 𝑥 two equals one. And 𝑦 two equals six. Note that it doesn’t matter which coordinate we have with the 𝑥 one, 𝑦 one values and which coordinate we have with the 𝑥 two, 𝑦 two values.
Starting with the distance formula then, we substitute in these values, giving us the square root of one minus negative two squared plus six minus negative seven squared. We must be very careful with our negative signs when we’re using this formula.
Simplifying then, we have the square root of three squared plus 13 squared, which is equal to the square root of nine plus 169, which then gives us the square root of 178. A decimal approximation for this square root is 13.34166 and so on.
For the purposes of comparing the side lengths here, an approximation of simply one decimal place is probably sufficient. At this point, we can already see that we have a length of eight and a length of 13.3, which means that we can already rule out an equilateral triangle.
Let’s clear some working space and find the length of our final line. This time, we can apply the distance formula between the coordinates of 𝐴 negative two, negative seven and 𝐶 at nine, six. Substituting in the designated 𝑥 one, 𝑦 one, 𝑥 two, and 𝑦 two values will give us the square root of nine minus negative two all squared plus six minus negative seven all squared.
And so our distance is equal to the square root of 11 squared plus 13 squared, which simplifies to the square root of 290. A decimal approximation to two decimal places here would be 17.03. We can therefore clearly see that this triangle does not have any equal sides. And so our answer is that this is a scalene triangle.