In triangle ABC, AB = 5 cm, BC = 12 cm, and AC = 13 cm. In triangle PQR, PQ = 12 cm, QR = 5 cm, and PR = 13 cm

Question

In triangle ABC, AB = 5 cm, BC = 12 cm, and AC = 13 cm. In triangle PQR, PQ = 12 cm, QR = 5 cm, and PR = 13 cm. Which of the following is a correct order of congruency?

Examoogle · Accepted Answer

The correct answer is Option B: Triangle ABC is congruent to Triangle QPR. Map the sides in order of increasing length: (5, 12, 13). In ABC, the sides are AB=5, BC=12, AC=13. In PQR, the sides are QR=5, PQ=12, PR=13. Matching the corresponding segments (AB=QR, BC=PQ, AC=PR) gives the mapping A-Q, B-R, C-P.

Step-by-Step Solution & Concepts:
Shortcut Trick: Map the sides in order of increasing length: (5, 12, 13). In ABC, the sides are AB=5, BC=12, AC=13. In PQR, the sides are QR=5, PQ=12, PR=13. Matching the corresponding segments (AB=QR, BC=PQ, AC=PR) gives the mapping A-Q, B-R, C-P.

Formula Used: $$	ext{SSS Congruence Principle: If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.}$$

Given: Triangle ABC has side lengths AB = 5 cm, BC = 12 cm, AC = 13 cm. Triangle PQR has side lengths PQ = 12 cm, QR = 5 cm, PR = 13 cm.

Steps: Identify side lengths of Triangle ABC: List the side lengths of triangle ABC in increasing order: $AB = 5$, $BC = 12$, $AC = 13$.. Identify side lengths of Triangle PQR: List the side lengths of triangle PQR: $QR = 5$, $PQ = 12$, $PR = 13$.. Map corresponding vertices: Match the vertices based on side equality: The side of length 5 is $AB$ in $\Delta ABC$ and $QR$ in $\Delta PQR$. The side of length 12 is $BC$ in $\Delta ABC$ and $PQ$ in $\Delta PQR$. This implies the correspondence $A \leftrightarrow Q$, $B \leftrightarrow R$, and $C \leftrightarrow P$.. Verify against options: The order of vertices that maintains the correspondence is $A 	o Q$, $B 	o R$, $C 	o P$. Thus, $\Delta ABC \cong \Delta QRP$.

Answer: B is correct because the correspondence of side lengths $AB=QR=5$, $BC=PQ=12$, and $AC=PR=13$ dictates that the vertices must be ordered as $A \leftrightarrow Q$, $B \leftrightarrow R$, and $C \leftrightarrow P$.

Exam Tip: This concept of mapping vertices by corresponding side lengths is identical to identifying transformations in Coordinate Geometry, where mapping $(x_1, y_1)$ to $(x_2, y_2)$ requires consistent ordering.

Examoogle · Answer

Option A: Triangle ABC is congruent to Triangle PQR

Examoogle · Answer

Option C: Triangle ABC is congruent to Triangle RQP

Examoogle · Answer

Option D: Triangle ABC is congruent to Triangle PRQ

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