Determine whether triangle PQR is congruent to triangle STU given the coordinates of the vertices. Explain. P(4,0),Q(2,-3)r(-1,4),S(-1,4), t(1,-1),u(4,6)
I would find the length of the lines PQ, PR, and QR
then find the lengths of the sides of the other triangle. They should be the same values
To determine whether triangle PQR is congruent to triangle STU, we need to compare their corresponding sides and angles.
1. Side PQ and side ST:
- The length of side PQ can be found using the distance formula:
PQ = sqrt((2 - 4)^2 + (-3 - 0)^2) = sqrt((-2)^2 + (-3)^2) = sqrt(4 + 9) = sqrt(13).
- The length of side ST can be found using the distance formula:
ST = sqrt((4 - 1)^2 + (6 - 4)^2) = sqrt((3)^2 + (2)^2) = sqrt(9 + 4) = sqrt(13).
- PQ = ST, so the corresponding sides are congruent.
2. Side QR and side TU:
- The length of side QR can be found using the distance formula:
QR = sqrt((-1 - 2)^2 + (4 + 3)^2) = sqrt((-3)^2 + 7^2) = sqrt(9 + 49) = sqrt(58).
- The length of side TU can be found using the distance formula:
TU = sqrt((1 - 4)^2 + (-1 - 6)^2) = sqrt((-3)^2 + (-7)^2) = sqrt(9 + 49) = sqrt(58).
- QR = TU, so the corresponding sides are congruent.
3. Side RP and side US:
- The length of side RP can be found using the distance formula:
RP = sqrt((-1 - 4)^2 + (4 - 0)^2) = sqrt((-5)^2 + 4^2) = sqrt(25 + 16) = sqrt(41).
- The length of side US can be found using the distance formula:
US = sqrt((4 - (-1))^2 + (6 - 4)^2) = sqrt((5)^2 + 2^2) = sqrt(25 + 4) = sqrt(29).
- RP is not equal to US, so the corresponding sides are not congruent.
Since all three pairs of corresponding sides of PQR and STU are not congruent, triangle PQR is not congruent to triangle STU.
To determine whether triangle PQR is congruent to triangle STU, we need to examine if their corresponding sides and angles are congruent.
1. Sides: Calculate the lengths of all three sides of both triangles PQR and STU using the distance formula:
Distance Formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For triangle PQR:
- Side PQ: d = sqrt((2 - 4)^2 + (-3 - 0)^2) = sqrt(4 + 9) = sqrt(13)
- Side QR: d = sqrt((-1 - 2)^2 + (4 - (-3))^2) = sqrt(9 + 49) = sqrt(58)
- Side RP: d = sqrt((-1 - 4)^2 + (4 - 0)^2) = sqrt(25 + 16) = sqrt(41)
For triangle STU:
- Side ST: d = sqrt((-1 - 1)^2 + (4 - (-1))^2) = sqrt(2^2 + 5^2) = sqrt(4 + 25) = sqrt(29)
- Side TU: d = sqrt((4 - (-1))^2 + (6 - (-1))^2) = sqrt(5^2 + 7^2) = sqrt(25 + 49) = sqrt(74)
- Side US: d = sqrt((4 - (-1))^2 + (6 - 4)^2) = sqrt(5^2 + 2^2) = sqrt(25 + 4) = sqrt(29)
2. Angles: Calculate the measures of all three angles of both triangles PQR and STU using the Law of Cosines:
Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C)
For triangle PQR:
- Angle P: cos(P) = (QP^2 + RP^2 - QR^2) / (2 * QP * RP)
- Angle Q: cos(Q) = (PQ^2 + QR^2 - RP^2) / (2 * PQ * QR)
- Angle R: cos(R) = (RP^2 + QR^2 - PQ^2) / (2 * RP * QR)
For triangle STU:
- Angle S: cos(S) = (TS^2 + US^2 - TU^2) / (2 * TS * US)
- Angle T: cos(T) = (ST^2 + TU^2 - US^2) / (2 * ST * TU)
- Angle U: cos(U) = (TU^2 + US^2 - ST^2) / (2 * US * TU)
Once you have calculated all the side lengths and angles for both triangles, compare them to check if they are congruent. If all corresponding sides and angles have the same values, then the triangles are congruent. If there is a mismatch in any side or angle, the triangles are not congruent.
Note: The calculations for this specific example were not provided, so it is up to you to perform the actual calculations.