How do I determine which two triangles on a graph are congruent. ∆QRS and ∆DEF or ∆ABC and ∆XYZ.

i found the coordinates of both sets:
QRS Q(-2,5), R(-2,1), S(-5, 1)
DEF D(5,-5), E(5,-2), F(2,-2)
or
ABC A(4,5), B(6,2), C(2,2)
XYZ X(-4,-5), Y(-6,-2), Z(-2,-2)

use the distance formula many times to find the lengths of the sides.

The two triangles with all congruent sides are congruent (SSS).

What is the distance formula?

http://www.google.com/#q=distance+formula

To determine if two triangles on a graph are congruent, you can use the side-lengths and angle measures. Congruent triangles must have the same side-lengths (in the same order) and the same angle measures.

Let's compare the two given sets of triangles:

Set 1: ∆QRS and ∆DEF
Coordinates:
QRS: Q(-2,5), R(-2,1), S(-5,1)
DEF: D(5,-5), E(5,-2), F(2,-2)

Set 2: ∆ABC and ∆XYZ
Coordinates:
ABC: A(4,5), B(6,2), C(2,2)
XYZ: X(-4,-5), Y(-6,-2), Z(-2,-2)

To determine if ∆QRS and ∆DEF are congruent, we need to check if their corresponding sides (QR and DE, RS and EF, QS and DF) have the same lengths.

1. Calculate the side lengths for both triangles:
∆QRS:
- QR = distance between Q and R = √((-2-(-2))^2 + (5-1)^2) = √(0 + 16) = √16 = 4
- RS = distance between R and S = √((-2-(-5))^2 + (1-1)^2) = √(3^2 + 0) = √9 = 3
- QS = distance between Q and S = √((-2-(-5))^2 + (5-1)^2) = √(3^2 + 16) = √25 = 5

∆DEF:
- DE = distance between D and E = √((5-5)^2 + (-5-(-2))^2) = √(0 + 9) = √9 = 3
- EF = distance between E and F = √((5-2)^2 + (-2-(-2))^2) = √(3^2 + 0) = √9 = 3
- DF = distance between D and F = √((5-2)^2 + (-5-(-2))^2) = √(3^2 + 9) = √18 = 3√2 (irrational length)

Comparing the side lengths, we can see that QR is congruent to DE, RS is congruent to EF, but QS is not congruent to DF. Therefore, ∆QRS is not congruent to ∆DEF.

Next, let's check if ∆ABC and ∆XYZ are congruent. Again, we compare the side lengths:

2. Calculate the side lengths for both triangles:
∆ABC:
- AB = distance between A and B = √((4-6)^2 + (5-2)^2) = √((-2)^2 + 9) = √4 + 9 = √13 (irrational length)
- BC = distance between B and C = √((6-2)^2 + (2-2)^2) = √(4^2 + 0) = √16 = 4
- AC = distance between A and C = √((4-2)^2 + (5-2)^2) = √(2^2 + 9) = √13 (irrational length)

∆XYZ:
- XY = distance between X and Y = √((-4-(-6))^2 + (-5-(-2))^2) = √((-2)^2 + 9) = √4 + 9 = √13 (irrational length)
- YZ = distance between Y and Z = √((-6-(-2))^2 + (-2-(-2))^2) = √((-4)^2 + 0) = √16 = 4
- XZ = distance between X and Z = √((-4-(-2))^2 + (-5-(-2))^2) = √((-2)^2 + 9) = √4 + 9 = √13 (irrational length)

Comparing the side lengths, we can see that AB is congruent to XY, BC is congruent to YZ, and AC is congruent to XZ. Therefore, ∆ABC is congruent to ∆XYZ.

In conclusion, the given triangles are congruent as follows: ∆ABC is congruent to ∆XYZ.

lol no brainer XD