Determine whether triangle PQR is congruent to triangle STU given the coordinates of the vertices. Explain your conclusion. coordinates of triangle PQR P(0,3) Q(0,-1) R(-2,-1) S(1,2) T(1,-2) U(-1,-2)
We have 2 rt triangles:
X1 = hor = X2 = 2. The hor. sides are
equal.
Y1 = ver = y2 = 4. The ver. sides are
equal.
Z1 = hyp. = Z2 = 4.5. The hyp are equal.
The corresponding sides are equal.
Therefore, the triangles are congruent.
CALCULATIONS:
Triangle # 1, PQR.
Y1 = 3 -(-1) = 3 + 1 = 4.
X1 = 0-(-2) = 2.
Triangle # 2,STU.
Y2 = 2 - (-2) = 2 + 2 = 4.
X2 = 1 - (-1) = 1 + 1 = 2.
To determine whether triangle PQR is congruent to triangle STU using the given coordinates of the vertices, we need to compare their corresponding sides and angles. Here's how you can do it:
1. Find the lengths of the sides of triangle PQR:
- Side PQ: The distance between points P(0,3) and Q(0,-1) can be found using the distance formula. The formula is:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the distance is sqrt((0 - 0)^2 + (-1 - 3)^2) = sqrt(0 + 16) = 4.
- Side QR: The distance between points Q(0,-1) and R(-2,-1) can be found using the distance formula. The formula is the same as above.
The distance is sqrt((-2 - 0)^2 + (-1 - (-1))^2) = sqrt(4 + 0) = 2.
- Side RP: The distance between points R(-2,-1) and P(0,3) using the distance formula.
The distance is sqrt((0 - (-2))^2 + (3 - (-1))^2) = sqrt(4 + 16) = sqrt(20) = 2*sqrt(5).
2. Find the lengths of the sides of triangle STU using the same approach:
- Side ST: The distance between points S(1,2) and T(1,-2).
- Side TU: The distance between points T(1,-2) and U(-1,-2).
- Side US: The distance between points U(-1,-2) and S(1,2).
3. Compare the lengths of the corresponding sides of the two triangles:
Match the sides of PQR with the sides of STU:
- Side PQ (4) with side ST (distance value).
- Side QR (2) with side TU (distance value).
- Side RP (2*sqrt(5)) with side US (distance value).
If the corresponding sides have the same lengths, then the triangles PQR and STU are congruent in terms of side lengths.
4. Additionally, you can also compare the angles between corresponding sides using slope. Find the slopes of lines PQ, QR, and RP in PQR, and the slopes of lines ST, TU, and US in STU. If the slopes are equal, then the corresponding angles are congruent.
By performing these comparisons and calculations, you can determine whether triangle PQR is congruent to triangle STU based on the given coordinates.
To determine whether triangle PQR is congruent to triangle STU, we can compare the lengths of the sides and the measures of the angles.
1. Length of PQ:
Using the distance formula, we calculate the length of PQ:
PQ = √[(x2 - x1)^2 + (y2 - y1)^2]
PQ = √[(0 - 0)^2 + (-1 - 3)^2]
PQ = √(0 + 16)
PQ = √16
PQ = 4
Now, let's calculate the length of ST:
ST = √[(x2 - x1)^2 + (y2 - y1)^2]
ST = √[(1 - 1)^2 + (-2 - 2)^2]
ST = √(0 + 16)
ST = √16
ST = 4
Therefore, PQ = ST.
2. Length of QR:
QR = √[(x2 - x1)^2 + (y2 - y1)^2]
QR = √[(-2 - 0)^2 + (-1 - (-1))^2]
QR = √[(-2)^2 + 0^2]
QR = √[4 + 0]
QR = √4
QR = 2
Now, let's calculate the length of TU:
TU = √[(x2 - x1)^2 + (y2 - y1)^2]
TU = √[(-1 - 1)^2 + (-2 - 2)^2]
TU = √[(-2)^2 + (-4)^2]
TU = √[4 + 16]
TU = √20
Therefore, QR ≠ TU.
Since the lengths of two corresponding sides PQ and ST are equal, but the lengths of the other two corresponding sides QR and TU are not equal, triangle PQR is not congruent to triangle STU.
Note: Keep in mind that this analysis only considers the side lengths. To determine congruence, you would also need to check if the corresponding angles are congruent.