Question

find the length of each side of the triangle to the nearest hundredth then find the perimeterThe length of AB is:The length of AD is:The length of BD is: The perimeter of the triangle is

Answer 1

Hello there. To solve this question, we have to remember some properties about coordinate geometry and the pythagorean theorem.

Given the following triangle in the xy plane:

To determine the length of the side AD, consider the following auxiliary triangle:

Notice it is a right triangle, that has hypotenuse with the same length as AD.

The legs of this triangle are given by the distance between each coordinate of the points A and D.

We find, by inspection, that the coordinates of A are (-5, 5) and D are (8, 3)

This means that the horizontal distance between A and D x coordinates are 13 and the height is 2.

The length is then given by the pythagorean theorem:

`\begin{gathered} L_(AD)^2=13^2+2^2=169+4=173 \\ \\ L_(AD)=√(173)\approx13.15 \\ \end{gathered}`

To determine AB, consider the next triangle:

B has coordinates (1, 1), so its horizontal distance to A is 6 and its vertical distance is 4.

The hypotenuse length is the same as the side AB, hence

`\begin{gathered} L_(AB)^2=6^2+4^2=36+16=52 \\ \\ L_(AB)=√(52)\approx7.21 \end{gathered}`

Finally, the side BD can also be calculated in the same manner:

The horizontal distance between B and D is 7 and the vertical distance is 2.

The length of BD is then given by

`\begin{gathered} L_(BD)^2=7^2+2^2=49+4=53 \\ \\ L_(BD)=√(53)\approx7.28 \\ \end{gathered}`

The perimeter is then given by the sum of the length of each side of the triangle:

`p\approx13.15+7.21+7.28=27.64`

These are the answers to this question.