Question

The points (1,−1), (9,−1), and (5,3) form a triangle. Use the dropdown menus and answer blanks below to prove the triangle is isosceles.

Answer 1

To prove that the triangle is isosceles, we need to show that two of its sides have the same length.

The formula to find the distance between two points (x1, y1) and (x2, y2) is given by:

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

Using this formula, we can find the length of each side of the triangle:

Distance between (1, -1) and (9, -1) = sqrt((9 - 1)^2 + (-1 - (-1))^2) = sqrt(8^2 + 0^2) = sqrt(64 + 0) = sqrt(64) = 8

Distance between (9, -1) and (5, 3) = sqrt((5 - 9)^2 + (3 - (-1))^2) = sqrt((-4)^2 + (4)^2) = sqrt(16 + 16) = sqrt(32)

Distance between (5, 3) and (1, -1) = sqrt((1 - 5)^2 + (-1 - 3)^2) = sqrt((-4)^2 + (-4)^2) = sqrt(16 + 16) = sqrt(32)

So, the lengths of the sides of the triangle are:

Side 1: 8
Side 2: sqrt(32)
Side 3: sqrt(32)

Since side 2 and side 3 have the same length (sqrt(32)), we can conclude that the triangle is isosceles.