Question

use Pythagorean theorem to find s,
a is -7,6
b is -2,-6
c is 10,-1

Answer 1

To use the Pythagorean theorem, we first need to determine the lengths of the sides of the triangle formed by the points \( A(-7, 6) \), \( B(-2, -6) \), and \( C(10, -1) \). We'll calculate the distances between each pair of points (sides of the triangle).

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

1. **Distance \( AB \)**:

\[
AB = \sqrt{((-2) - (-7))^2 + ((-6) - 6)^2}
\]
\[
= \sqrt{(5)^2 + (-12)^2}
\]
\[
= \sqrt{25 + 144}
\]
\[
= \sqrt{169}
\]
\[
= 13
\]

2. **Distance \( BC \)**:

\[
BC = \sqrt{(10 - (-2))^2 + ((-1) - (-6))^2}
\]
\[
= \sqrt{(12)^2 + (5)^2}
\]
\[
= \sqrt{144 + 25}
\]
\[
= \sqrt{169}
\]
\[
= 13
\]

3. **Distance \( AC \)**:

\[
AC = \sqrt{(10 - (-7))^2 + ((-1) - 6)^2}
\]
\[
= \sqrt{(17)^2 + (-7)^2}
\]
\[
= \sqrt{289 + 49}
\]
\[
= \sqrt{338}
\]

Now that we have the lengths of the sides of the triangle:

- \( AB = 13 \)
- \( BC = 13 \)
- \( AC = \sqrt{338} \)

To determine if this triangle satisfies the Pythagorean theorem \( a^2 + b^2 = c^2 \), we can square the lengths:

Let \( c^2 \) be the longest side (in this case, \( AC \)):

1. \[ a^2 + b^2 = AB^2 + BC^2 = 13^2 + 13^2 = 169 + 169 = 338 \]
2. \[ c^2 = AC^2 = (\sqrt{338})^2 = 338 \]

Since \( a^2 + b^2 = c^2 \):

\[
338 = 338
\]

The triangle with vertices \( A \), \( B \), and \( C \) is a right triangle, with \( AC \) being the hypotenuse. The value \( s \) is not explicitly defined, but if you wanted to calculate the semi-perimeter or any specific value related to the triangle, please specify.