Question

Given points A(−2,0), B(−5,3), C(−9,−1), P(7,6), Q(4,0), and R(−4,4), which of the following proves that △ABC~△PQR?

Answer options:
By the Distance Formula,AB=32–√, BC=42–√, and CA=52–√.Also, PQ=35–√, QR=45–√, and RP=55–√.Therefore, ABPQ=BCQR=CARP=2√5√=10√5, and therefore, △ABC∼△PQR by SAS ∼.

By the Distance Formula,AB=50, BC=32, and CA=18.Also, PQ=125, QR=90, and RP=45.Therefore, ABPQ=BCQR=CARP=25, and therefore, △ABC∼△PQR by SSS ∼.

By the Distance Formula,AB=18, BC=32, and CA=50.Also, PQ=45, QR=90, and RP=125.Therefore, ABPQ=BCQR=CARP=25, andtherefore, △ABC∼△PQR by SAS ∼.

By the Distance Formula,AB=32–√, BC=42–√, and CA=52–√.Also, PQ=35–√ QR=45–√, and RP=55–√.Therefore, ABPQ=BCQR=CARP=2√5√=10√5,and therefore, △ABC∼△PQR by SSS ∼.

Answer 1

Based on the SSS similarity theorem, △ABC ~ △PQR because AB/PQ = BC/QR = CA/RP = √2/√5 = √10/5 (option D).

The SSS Similarity Theorem

Two triangles having three pairs of sides that are proportional can be proven to be similar by the SSS similarity theorem.

If the triangle ABC and triangle PQR are similar, their corresponding sides will be proportional, meaning that: AB/PQ = BC/QR = CA/RP.

Therefore, using the distance formula, $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, the sides of each triangle is found.

Therefore, it shows that:

AB/PQ = BC/QR = CA/RP = √2/√5 = √10/5

Therefore, based on the SSS similarity theorem, △ABC ~ △PQR because AB/PQ = BC/QR = CA/RP = √2/√5 = √10/5 (option D).

Learn more about the SSS similarity theorem on:

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