Questions: (1,-5),(-3,-6) and (5,-6) Consider the following points. Step 1 of 2: Determine whether or not the given points form a right triangle. If the triangle is not a right triangle, determine if it is isosceles or scalene. Answer Right Triangle Isosceles Triangle Scalene Triangle

Solution Steps

To determine the type of triangle formed by the given points, we can calculate the distances between each pair of points to find the lengths of the sides of the triangle. Then, we check if the triangle is a right triangle by verifying if the Pythagorean theorem holds for any combination of the sides. If not, we check if two sides are equal to determine if it is isosceles, otherwise, it is scalene.

We have the points \( A = (1, -5) \), \( B = (-3, -6) \), and \( C = (5, -6) \). The lengths of the sides are calculated as follows:

\[ AB = \sqrt{((-3) - 1)^2 + ((-6) - (-5))^2} = \sqrt{(-4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.1231 \]

\[ BC = \sqrt{(5 - (-3))^2 + ((-6) - (-6))^2} = \sqrt{(8)^2 + (0)^2} = \sqrt{64} = 8.0 \]

\[ CA = \sqrt{(5 - 1)^2 + ((-6) - (-5))^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.1231 \]

The lengths of the sides are approximately:

\[ AB \approx 4.1231, \quad BC = 8.0, \quad CA \approx 4.1231 \]

Sorting these values gives us:

\[ sides = [4.1231, 4.1231, 8.0] \]

To check if the triangle is a right triangle, we apply the Pythagorean theorem:

\[ (4.1231)^2 + (4.1231)^2 \approx 16.9999 \quad \text{and} \quad (8.0)^2 = 64 \]

Since \( 16.9999 \neq 64 \), the triangle is not a right triangle.

Next, we check for isosceles properties:

\[ AB \approx CA \quad \text{(both are approximately } 4.1231\text{)} \]

Since two sides are equal, the triangle is isosceles.

Final Answer