### Video Transcript

What is the kind of triangle that the points ๐ด negative two, negative seven; ๐ต one, six; and ๐ถ nine, six form with respect to its sides?

Letโs 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 weโll need to work out the length of each side and compare.

Starting with the simplest horizontal length between ๐ต and ๐ถ, we know that since it goes from the coordinate ๐ต with ๐ฅ-value one to the coordinate ๐ถ with ๐ฅ-value nine. Then the length must be eight units. It might be clear at this point that weโre unlikely to have any other length of eight and that the other two lengths probably arenโt equal. But letโs 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 ๐ฅ one, ๐ฆ one and ๐ฅ two, ๐ฆ two is equal to the square root of ๐ฅ two minus ๐ฅ one all squared plus ๐ฆ two minus ๐ฆ one all squared. Starting with the length of line ๐ด๐ต, we have ๐ฅ one equals negative two. ๐ฆ one equals negative seven. ๐ฅ two equals one. And ๐ฆ two equals six. Note that it doesnโt matter which coordinate we have with the ๐ฅ one, ๐ฆ one values and which coordinate we have with the ๐ฅ two, ๐ฆ 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 weโre 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.

Letโs clear some working space and find the length of our final line. This time, we can apply the distance formula between the coordinates of ๐ด negative two, negative seven and ๐ถ at nine, six. Substituting in the designated ๐ฅ one, ๐ฆ one, ๐ฅ two, and ๐ฆ 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.