Reiny ..thanks for the formula and example. I tried it and i get that both sets of triangles are congruent but
it is multiple choice with one answer. I don't know what i am doing wrong.
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,-6), 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)
Let's see what's what here...
QR = √((-2+2)^2 + (1-5)^2) = √16
RS = √((-5+2)^2 + (1-1)^2) = √9
SQ = √((-2+5)^2 + (5-1)^2) = √25
DE = √((5-5)^2 + (-2+6)^2) = √16
EF = √((2-5)^2 + (-2+2)^2) = √9
FD = √((5-2)^2 + (-6+2)^2) = √25
So, QRS ≅ DEF
AB = √((6-4)^2 + (2-5)^2) = √13
BC = √((2-6)^2 + (2-2)^2) = √16
CA = √((4-2)^2 + (5-2)^2) = √13
XY = √((-6+4)^2 + (2+5)^2) = √13
YZ = √((-2+6)^2 + (-2+2)^2) = √16
ZX = √((-4+2)^2 + (-5+2)^2) = √13
So, ABC ≅ XYZ
So, assuming there are no typos, you are correct and the answer key is wrong.
In this case QRS and DEF are easy, since they are right triangles with legs parallel to the axes, and are easy to measure.
ABC and XYZ are just reflected through the origin (the coordinates have just changed signs), so they also must be congruent.
Ok..great..thx for checking.
To determine which two triangles on a graph are congruent, you need to compare the sides and angles of both triangles. Here's how you can do it step-by-step:
1. Compare the side lengths:
- Measure the length of all three sides of each triangle.
- For example, for ∆QRS, measure the distance between Q and R, between R and S, and between S and Q. Repeat this for ∆DEF, ∆ABC, and ∆XYZ.
- Use the distance formula: distance = sqrt( (x2 - x1)^2 + (y2 - y1)^2 ).
2. Compare the angles:
- Measure the angles of each triangle using a protractor or by using trigonometric functions (sin, cos, tan).
- For example, for ∆QRS, measure the angles ∠Q, ∠R, and ∠S. Repeat this for ∆DEF, ∆ABC, and ∆XYZ.
3. Compare the measurements:
- Compare the side lengths of ∆QRS and ∆DEF. Are they equal? If all three sides of one triangle are equal to the three sides of the other triangle, then they are congruent in terms of side lengths.
- Check the angles of ∆QRS and ∆DEF. Are they equal? If all three angles of one triangle are equal to the three angles of the other triangle, then they are congruent in terms of angles.
4. Repeat the same process for ∆ABC and ∆XYZ:
- Compare the side lengths and angles of ∆ABC and ∆XYZ using the same steps outlined above.
By comparing the measurements of sides and angles for both sets of triangles, you should be able to determine which two triangles are congruent. Remember, for triangles to be congruent, their corresponding sides and angles must be equal.