Question

rowan, ari, and arthur are playing catch in a field. rowan is located at point a (52,40), ari is located at point b (-39,39), and arthur is located at point c (91,79). if the f
right triangle
equilateral triangle
scalene triangle
isosceles triangle

Explanation:

To determine the type of triangle formed by points \( A(52, 40) \), \( B(-39, 39) \), and \( C(91, 79) \), we use the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) to find the lengths of the sides \( AB \), \( BC \), and \( CA \).

Step 1: Calculate the length of \( AB \)

For points \( A(52, 40) \) and \( B(-39, 39) \):

$$
LATEXBLOCK0
$$

Step 2: Calculate the length of \( BC \)

For points \( B(-39, 39) \) and \( C(91, 79) \):

$$
LATEXBLOCK1
$$

Step 3: Calculate the length of \( CA \)

For points \( C(91, 79) \) and \( A(52, 40) \):

$$
LATEXBLOCK2
$$

Now, we compare the lengths:

• \( AB = \sqrt{8282} \approx 91.0 \)
• \( BC = \sqrt{18500} \approx 136.0 \)
• \( CA = \sqrt{3042} \approx 55.15 \)

Since all three side lengths are different, the triangle is scalene. The initially selected option "Equilateral Triangle" is incorrect because an equilateral triangle requires all sides to be equal, which is not the case here. A scalene triangle has all sides of different lengths, which matches our calculations.

Answer:

Scalene Triangle