Geometry Homework!!!! ARRRRGGGGGGHHHHHHHH!!!!!?

Question

Verify that triangle PQR is a right triangle. P(3,2) , Q(-3,6) , R(5,5) You are supposed to use Distance Formulas and Pythagorean Theorem and I missed this class! HELP!

Answer 1

PQ ² = ( - 3 - 3 ) ² + (6 - 2) ² = 52
QR ² = (5 + 3) ² + (5 - 6) ² = 65
PR ² = (5 - 3) ² + (5 - 2) ² = 13
PQ ² + PR ² = QR ²
Thus ΔPQR is right angled at P

Answer 2

It's probably easier to do by slopes...but that avoids what you are supposed to learn.

Do you know the Pythagorean Theorem? It says that if a right triangle has legs A and B and hypotenuse C, then A^2 + B^2 = C^2. In order to show that PQR is a right triangle, you actually need the converse of the Pythagorean Theorem, which says that if the equation works out, then the triangle is right.

Let's take your problem. Consider P and Q. Draw a third point T(3,6), which has the same x-coordinate as P and the same y-coordinate as Q. Can you see why PQT is a right triangle? How long are the legs PT and QT? Can you use the theorem to find the length of PQ?

PT=4, because P and T have the same x-coordinate but differ by 4 in the y-coordinate.

QT=6, because Q and T have the same y-coordinate, but differ by 6 in the x-coordinate.

Since the right angle is at T, PQ is the hypotenuse, and:
(PT)^2 + (QT)^2 = (PQ)^2
4^2 + 6^2 = (PQ)^2
16 + 36 = (PQ)^2
(PQ)^2 = 52
PQ = Sqrt(52) = 2Sqrt(13).

(For the rest of your problem, it will be more helpful to know (PQ)^2 than PQ.)

For points Q and R, try to find a third point U that acts the same way T did.

Repeat for P and R.

Do the two smaller squares add up to the larger square?

Answer 3

Your mathematical intuition is correct...need to work the distance formulas for all three sides and use the Pythagoras theorem to confirm right angle condition.

D (QP) = sqrt[(3+3)^2 + (2-6)^2] = sqrt(36 + 16) = sqrt(52)
D(PR) = sqrt[(3-5)^2 + (2-5)^2] = sqrt(4 + 9) = sqrt(13)
D(QR) = sqrt[(5+3)^2 + (5-6)^2] = sqrt(64 + 1) = sqrt(65)

D(QP)^2 + D(PR)^2 = D(QR)^2

52 + 13 = 65

65 = 65

Right Triangle

and that's how is done!