Question

You are given two of the three points that make a triangle, A(2,4) and B(3,-1). Determine which the points below will make the triangle isosceles and right. Responses (-3,4) (-3,4) (-4,3) (-4,3) (-5,3) (-5,3) (-3,3)

Answer 1

To determine which point will make the triangle isosceles and right, we can use the distance formula to find the length of AB and the slopes of AB and the segment between each point and one of the known points A and B.

The distance formula is given by:

D = sqrt((x2 - x1)^2 + (y2 - y1)^2).

First, let's find the length of AB:

AB = sqrt((3 - 2)^2 + (-1 - 4)^2)

= sqrt(1 + 25)

= sqrt(26)

The slope of AB is given by:

mAB = (y2 - y1) / (x2 - x1)

= (-1 - 4) / (3 - 2)

= -5 / 1

= -5

Now, let's calculate the slopes between the points and A, and the points and B:

For the point C (-3, 4):

mAC = (4 - 4) / (-3 - 2)

= 0 / -5

= 0

So, the slope of AC is 0.

For the point D (-4, 3):

mAD = (3 - 4) / (-4 - 2)

= -1 / -6

= 1/6

So, the slope of AD is 1/6.

For the point E (-5, 3):

mAE = (3 - 4) / (-5 - 2)

= -1 / -7

= 1/7

So, the slope of AE is 1/7.

For the point F (-3, 3):

mBF = (3 - 4) / (-3 - 3)

= -1 / -6

= 1/6

So, the slope of BF is 1/6.

For the point G (-4, 3):

mBG = (3 - 4) / (-4 - 3)

= -1 / -7

= 1/7

So, the slope of BG is 1/7.

Based on these calculations, points D and E will make the triangle isosceles, and point D will make the triangle right, as it has a slope perpendicular to AB.