Congruence Check for Triangles PQR and STU
To determine if triangle PQR and triangle STU are congruent, we calculate the lengths of their sides using the coordinates provided.
Triangle PQR Side Lengths
Vertices are P(1, 1), Q(4, 1), and R(1, 5).
- Side PQ: The y-coordinates are the same (1). The length is the absolute difference of the x-coordinates: $|4 - 1| = 3$.
- Side PR: The x-coordinates are the same (1). The length is the absolute difference of the y-coordinates: $|5 - 1| = 4$.
- Side QR: Using the distance formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$: $QR = \sqrt{(1-4)^2 + (5-1)^2} = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
The side lengths of triangle PQR are 3, 4, and 5.
Triangle STU Side Lengths
Vertices are S(6, 2), T(9, 2), and U(6, 6).
- Side ST: The y-coordinates are the same (2). The length is the absolute difference of the x-coordinates: $|9 - 6| = 3$.
- Side SU: The x-coordinates are the same (6). The length is the absolute difference of the y-coordinates: $|6 - 2| = 4$.
- Side TU: Using the distance formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$: $TU = \sqrt{(6-9)^2 + (6-2)^2} = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
The side lengths of triangle STU are 3, 4, and 5.
Congruence Conclusion
Comparing the side lengths:
- PQ = ST = 3
- PR = SU = 4
- QR = TU = 5
Since all three corresponding sides of triangle PQR are equal in length to the three corresponding sides of triangle STU, the triangles are congruent by the Side-Side-Side (SSS) congruence postulate.
Similar Questions
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Important Questions from Triangles, Congruence and Similarity
Angle between the internal bisectors of two angles ∠B and ∠C of a ΔABC is 132°, then the value of ∠A is
In ΔPQR, PQ = PR and S is a point on QR such that ∠PSQ = 96° + ∠QPS and ∠QPR = 132°. What is the measure of ∠PSR?
In Δ ABC, ∠A = 50°. If the bisectors of the angle B and angle C, meet at a point O, then ∠BOC is equal to:
Triangle ABC is right angled at B. BD is an altitude intersecting AC at D. If AC = 9 cm and CD = 3 cm. then find the measure of AB (in cm).
The base and altitude of an isosceles triangle are 10 cm and 12 cm respectively. Then the length of each equal side is:
