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QuestionCheck if the triangle with vertices B(-1,5), A(2,3), and C(0,0) is a right triangle using the distances.

Studdy Solution

STEP 1

Assumptions1. The vertices of the triangle are B (-1,5), A (,3), and C (0,0). . We are to determine if the triangle is a right triangle.

STEP 2

To determine if a triangle is a right triangle, we can use the Pythagorean theorem. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. We can write this asc2=a2+b2c^2 = a^2 + b^2where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

STEP 3

First, we need to find the lengths of the sides of the triangle. We can do this by using the distance formulad=(x2−x1)2+(y2−y1)2d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}where (x1, y1) and (x2, y2) are the coordinates of two points.

STEP 4

Let's calculate the length of side AB. Plug in the coordinates of points A and B into the distance formulaAB=(2−(−1))2+(3−)2AB = \sqrt{(2 - (-1))^2 + (3 -)^2}

STEP 5

Calculate the length of side AB.
AB=(2−(−1))2+(3−5)2=9=3AB = \sqrt{(2 - (-1))^2 + (3 -5)^2} = \sqrt{9} =3

STEP 6

Now, calculate the length of side BC. Plug in the coordinates of points B and C into the distance formulaBC=((−1)−0)2+(5−0)2BC = \sqrt{((-1) -0)^2 + (5 -0)^2}

STEP 7

Calculate the length of side BC.
BC=((−1)−0)2+(5−0)2=26BC = \sqrt{((-1) -0)^2 + (5 -0)^2} = \sqrt{26}

STEP 8

Finally, calculate the length of side AC. Plug in the coordinates of points A and C into the distance formulaAC=(2−0)2+(3−0)2AC = \sqrt{(2 -0)^2 + (3 -0)^2}

STEP 9

Calculate the length of side AC.
AC=(2−)2+(3−)2=13AC = \sqrt{(2 -)^2 + (3 -)^2} = \sqrt{13}

STEP 10

Now, we can check if the triangle is a right triangle by using the Pythagorean theorem. The longest side, BC, should be the hypotenuse. So, we should check ifBC2=AB2+AC2BC^2 = AB^2 + AC^2

STEP 11

Plug in the lengths of the sides into the equation\sqrt{26}^ =3^ + \sqrt{13}^

STEP 12

Calculate the left and right sides of the equation26=9+26 =9 +The equation is not true, so the triangle is not a right triangle.

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