Question

Find the length of each side of the triangle determined by the three points and state whether the triangle is an isosceles​ triangle, a right​ triangle, neither of​ these, or both.​ (An isosceles triangle is one in which at least two of the sides are of equal​ length.) Upper P 1 equals left parenthesis negative 2 comma negative 5 right parenthesisP1=(−2,−5)​, Upper P 2 equals left parenthesis 0 comma 15 right parenthesis commaP2=(0,15), Upper P 3 equals left parenthesis 9 comma 4 right parenthesisP3=(9,4)

Answer 1

Answer: Both isosceles and right triangle

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Step-by-step explanation

I'll label the 3 points as A,B,C.

A = (-2,-5)

B = (0,15)

C = (9,4)

Let's find the distance from A to B. It's the same as finding the length of segment AB.

`(x_1,y_1) = (-2,-5) \text{ and } (x_2, y_2) = (0,15)\\\\d = √((x_1 - x_2)^2 + (y_1 - y_2)^2)\\\\d = √((-2-0)^2 + (-5-15)^2)\\\\d = √((-2)^2 + (-20)^2)\\\\d = √(4 + 400)\\\\d = √(404)\\\\d = √(4*101)\\\\d = √(4)*√(101)\\\\d = 2√(101)\\\\d \approx 20.099751\\\\`

Segment AB is exactly
`2√(101)` units long.

Segment AB is approximately 20.099751 units long.

If you follow similar steps then you'll find that segment BC is
`√(202) \approx 14.21267` units long. The same will apply to segment AC.

Therefore, BC = AC and it proves the triangle is isosceles

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To prove if we have a right triangle or not, we'll need to find the slopes of AB, BC and AC

I'll show the steps how to find the slope of line through A(-2,-5) and B(0,15).

`(x_1,y_1) = (-2,-5) \text{ and } (x_2,y_2) = (0,15)\\\\m = \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}\\\\m = \frac{\text{y}_(2) - \text{y}_(1)}{\text{x}_(2) - \text{x}_(1)}\\\\m = (15 - (-5))/(0 - (-2))\\\\m = (15 + 5)/(0 + 2)\\\\m = (20)/(2)\\\\m = 10\\\\`

Line AB has slope 10.

I'll skip steps, but line BC has slope -11/9

While slope AC is 9/11

The slopes of BC and AC are negative reciprocals of one another. They multiply to -1; which means AC and BC are perpendicular. Therefore, triangle ABC is a right triangle.

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Conclusion: Triangle ABC is an isosceles right triangle