Question

. Do the points P (-2, -3), Q (4, 1) and R (2, 4) form a right triangle?

Answer 1

Answer:

Since the length of the sides respects the Pythagorean theorem, these points form a right triangle.

Step-by-step explanation:

Distance between two points:

Suppose that we have two points, $(x_1,y_1)$ and $(x_2,y_2)$. The distance between them is given by:

$D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Right triangle:

Sum of the squares of the two smaller sides is equal to the square of the largest side(Pythagorean theorem).

Length of side PQ:

P (-2, -3), Q (4, 1)

$a = \sqrt{(4 - (-2))^2+(1 - (-3))^2} = \sqrt{6^2 + 4^2} = \sqrt{52}$

Length of side PR:

P (-2, -3), R (2,4)

$b = \sqrt{(2 - (-2))^2+(4 - (-3))^2} = \sqrt{4^2 + 7^2} = \sqrt{65}$

Length of side QR

Q (4, 1), R (2,4)

$c = \sqrt{(2 - 4)^2+(4 - 1)^2} = \sqrt{2^2 + 3^2} = \sqrt{13}$

Pythagorean Theorem:

Smaller sides: a and c

Largest side: b

So

$a^2 + c^2 = b^2$

$(\sqrt{52})^2 + (\sqrt{13})^2 = (\sqrt{65})^2$

$52 + 13 = 65$

$65 = 65$

Since the length of the sides respects the Pythagorean theorem, these points form a right triangle.