### Video Transcript

A triangle has vertices at the points ๐ด, ๐ต, and ๐ถ with the coordinates three, three; negative one, three; and seven, negative six, respectively. Work out the perimeter of triangle ๐ด๐ต๐ถ. Give your answer solution to two decimal places.

Weโre trying to work out the perimeter of this triangle. This means we need to add the distance from ๐ด๐ต to the distance from ๐ต to ๐ถ to the distance from ๐ถ to ๐ด, side one of the triangle plus side two of the triangle plus side three of the triangle. Together theyโll equal the perimeter. But to do that, weโll need to know how to find the distance between two points. And we have a formula for calculating the distance between two points. The distance between two points is equal to the square root of ๐ฅ two minus ๐ฅ one squared plus ๐ฆ two minus ๐ฆ one squared. We know that ๐ด is located at three, three. And ๐ต is located at negative one, three, while ๐ถ is found at seven, negative. Six. If we want to find the distance from ๐ด to ๐ต, we can let ๐ด be ๐ฅ one, ๐ฆ one and ๐ต be ๐ฅ two, ๐ฆ two. ๐ฅ two minus ๐ฅ one is negative one minus three. Then square that value. ๐ฆ two minus ๐ฆ one, in this case, would be three minus three. The ๐ฆ-coordinates of both ๐ด and ๐ต are three.

Negative one minus three equals negative four. We need to square that value. Three minus three equals zero. And zero squared equals zero. And that means we can drop that term. We need to square negative four which is 16 and the square root of 16 is four. This means the distance from point ๐ด to point ๐ต in this triangle is four units. Weโll follow the same procedure to find the distance from ๐ต to ๐ถ. In this case, we label ๐ต as ๐ฅ one, ๐ฆ one and ๐ถ as ๐ฅ two, ๐ฆ two. After we plug in our values, we have the square root of seven minus negative one squared plus negative six minus three squared. Seven minus negative one is seven plus one, so we have eight squared. Negative six minus three is negative nine, so we have negative nine squared.

Weโll take the square root of eight squared plus negative nine squared. Eight squared is 64. Nine squared, or negative nine squared, is 81. Now, weโll need to add 64 and 81 which is 145. Line ๐ต๐ถ has a length of the square root of 145. We will round our answer solution to two decimal places. But for now, weโll keep it as the square root of 145 and round in our final step. We now need to find the distance from ๐ด to ๐ถ or from ๐ถ to ๐ด. Either way will work as long as youโre consistent in youโre labeling. Weโll let ๐ด be ๐ฅ one, ๐ฆ one and ๐ถ be ๐ฅ two, ๐ฆ two. The distance from ๐ด to ๐ถ will be equal to the square root of seven minus three squared plus negative six minus three squared. Seven minus three is four, so weโll have four squared. Negative six minus three is negative nine. We have four squared plus negative nine squared. Negative nine squared equals 81. Four squared equals 16. And so we can say that the distance from ๐ด to ๐ถ is the square root of 97. 16 plus 81 is 97.

At this point, we could find the perimeter. The perimeter equals four plus the square root of 145 plus the square root of 97. This method, using the distance between two pointsโ formula does not require us to sketch a picture. However, sometimes itโs helpful to use a coordinate plane to see whatโs going on. If we sketch a coordinate plane, we can plot the three points of our triangle ๐ด located at three, three; ๐ต located at negative one, three; and ๐ถ located at seven, negative six. From there, we need to connect the three points, ๐ด๐ต, ๐ต๐ถ, and ๐ด๐ถ. By sketching the graph of this triangle, we can find the distance from ๐ด to ๐ต visually. ๐ด to ๐ต is four units because they fall on the same horizontal line. Instead of doing the long calculation, we could have just inspected to see that it was four units.

Now, we canโt do that kind of inspection, even on the graph from ๐ด to ๐ถ or from ๐ต to ๐ถ. But thereโs something else we can do here. We can create a right triangle that has a high hypotenuse of ๐ต๐ถ. The vertical distance from ๐ต to ๐ถ is from negative six to three. And thatโs nine units. The horizontal distance from ๐ต to ๐ถ is from negative one to seven, which is eight units. To find the length ๐ต๐ถ, we could say that ๐ต๐ถ squared equals nine squared plus eight squared. Weโre using the Pythagorean theorem to find the length of ๐ต๐ถ. Nine squared is 81; eight squared is 64; ๐ต๐ถ squared equals 145. We take the square root of both sides and we see that ๐ต๐ถ equals the square root of 145. And what youโre seeing is that the distance formula is actually the Pythagorean theorem rearranged.

The distance formula takes the vertical distance and squares it, the horizontal distance and squares it, and then takes the square root. We could also do this to find the distance from ๐ด to ๐ถ. The distance from ๐ด to ๐ถ squared will be equal to the horizontal distance of four, four squared, plus the vertical distance of nine squared. Again weโll have 16 plus 81. ๐ด๐ถ squared equals 97, and if we take the square root of both sides, we confirm that ๐ด๐ถ does in fact equal the square root of 97. Back to our original goal of finding the perimeter, if we add four plus the square root of 145 plus the square root of 97, our calculator will give us something like 28.890452 continuing. Weโre rounding to two decimal places to the hundredths place we look to the digit to the right of the hundredths place to do our rounding, which in this case is a zero. So weโll round down. And weโll say that the perimeter of this triangle is 28.89 units, 28 and 89 hundredths.