Question

Triangle CAT has vertices at C (-6,0), A (4,-2), and T (5,3). What type of triangle is CAT? Show your work.

a. obtuse
b. isosceles
c. equilateral
d. right

Answer 1

By calculating the side lengths of triangle CAT using the coordinates of the vertices and applying the Pythagorean theorem, it is determined that triangle CAT is a right triangle.

Step-by-step explanation:

To determine the type of triangle CAT, we need to calculate the lengths of its sides using the coordinates of its vertices. We will then apply the Pythagorean theorem to see if it satisfies the condition for a particular type of triangle.

First, we find the lengths of the sides:

• Side CA: √[(4 - (-6))2 + (-2 - 0)2] = √[102 + 22] = √104
• Side AT: √[(5 - 4)2 + (3 - (-2))2] = √[12 + 52] = √26
• Side CT: √[(5 - (-6))2 + (3 - 0)2] = √[112 + 32] = √130

We then compare these lengths using the Pythagorean theorem:

For a right triangle, the sum of the squares of the lengths of the two shorter sides should equal the square of the length of the longest side. We check if CA2 + AT2 = CT2:

• CA2 = 104
• AT2 = 26
• CT2 = 130

Calculating the sum of the squares of the two shorter sides: 104 + 26 = 130. This is equal to CT2. Therefore, triangle CAT is a right triangle.