A triangle XYZ has its vertices at X(2,4), Y(6,-4), and Z(-4,1) What is the perimeter of the triangle XYZ?

Answer 1

Hi Karla. For each side, you need to use the distance formula.

For side XY, we can make (2,4) X1, Y1. We can also make (6, -4) X2, Y2. Using the formula, it’s the square root of 6 - -2 squared + -4 -4 squared. They give us the square root of 8 squared + -8 squared, = the square root of 128 squared, or 8 square root of 2.

For side YZ, we can make (6, -4) X1, Y1. We can also make (-4, 1) X2, Y2. Using the formula, it’s the square root of -4 -6 squared + 1- -4 squared. That gives you the square root of -10 squared + -3 squared, = the square root of 109 squared. That can’t be simplified.

For side XZ, we can make (2,4) X1, Y1. We can also make (-4,1) X2, Y2. Using the formula, it’s the square root of -4 -2 squared, + 1 -4 squared. You get the square root of -6 squared + -3 squared = the square root of 45, = 3 square root of 5.

So the perimeter is 8 square root of 2 + the square root of 109 + 3 square root of 5.

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Answer 2

The perimeter of a figure is the sum of all the sides. So you must find the distance between each set of points, then add them together. Use the distance formula: d = √[ (x₁ - x₂)² + (y₁ - y₂)²]

a) Find the distance between X(2, 4) and Y(6, -4)

d = √[ (6 - 2)² + (-4 - 4)²] = √[ (4)² + (-8)²] = √ (16 + 64) = √80

b) Find the distance between X(2, 4) and Z(-4, 1)

d = √[ (-4 - 2)² + (1 - 4)²] = √[ (-6)² + (-3)²] = √ (36 + 9) = √45

c) Find the distance between Y(6, -4) and Z(-4, 1)

d = √[ (-4 - 6)² + (1 - (-4)²] = √[ (-10)² + (5)²] = √ (100 + 25) = √125

d) The perimeter of the triangle XYZ is step a + step b + step c

P = √80 + √45 + √125 = 16√5 + 3√5 + 5√5 = 24√5

Answer 3

Hi Karla! Solving your question basically just requires using the distance formula three times. As a quick reminder, the distance formula is d = sqrt[(x₂-x₁)² - (y₂-y₁)²] for any two points (x₁,y₁) and (x₂,y₂). You will have to use the formula three times (from X to Y, from Y to Z, and from Z to X) to get the lengths of the three sides of the triangle. Once you do that, you find the perimeter by just adding those three sidelengths together.

I hope that helps answers your question Karla. Feel free to ask if you have any other questions. Have a lovely day!