How do I prove that these are the vertices of an isosceles triangle: (-3,0), (0,4), (3,0)?

Question

Theodore Abad · Accepted Answer

Just determine the three sides.

#A = (-3, 0)# on x axis #B = (0, 4)# on y axis #C = (3, 0)# on x axis #AC = 6# #AB^2 = 3^2 + 4^2# #BC^2 = 3^2 + 4^2# #AB = BC = 5#

Parker Allen · Answer

undefined To prove that the points (-3,0), (0,4), and (3,0) are vertices of an isosceles triangle, we need to show that the distances between these points satisfy the definition of an isosceles triangle, which means that at least two sides of the triangle are of equal length.

Calculate the distances between each pair of points:

1. Distance between (-3,0) and (0,4):
$$ d_1 = \sqrt{(0 - (-3))^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

2. Distance between (0,4) and (3,0):
$$ d_2 = \sqrt{(3 - 0)^2 + (0 - 4)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

3. Distance between (-3,0) and (3,0):
$$ d_3 = \sqrt{(3 - (-3))^2 + (0 - 0)^2} = \sqrt{6^2 + 0} = \sqrt{36} = 6 $$

We can see that $d_1 = d_2 = 5$, while $d_3 = 6$. Since two sides of the triangle have equal lengths, namely $d_1$ and $d_2$, the triangle formed by these points is indeed an isosceles triangle.

Savannah Aguilar · Answer

undefined To prove that the given points (-3,0), (0,4), and (3,0) are vertices of an isosceles triangle, we need to show that the distances between certain pairs of points are equal.

1. Calculate the distances between each pair of points:

- Distance between (-3,0) and (0,4):
$$ d_1 = \sqrt{(0 - (-3))^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

- Distance between (0,4) and (3,0):
$$ d_2 = \sqrt{(3 - 0)^2 + (0 - 4)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

- Distance between (-3,0) and (3,0):
$$ d_3 = \sqrt{(3 - (-3))^2 + (0 - 0)^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6 $$

2. Since $ d_1 = d_2 = 5 $ and $ d_3 = 6 $, we can see that two sides of the triangle have equal lengths, making it an isosceles triangle.

Therefore, the given points (-3,0), (0,4), and (3,0) are vertices of an isosceles triangle.