Suppose BC is a given line segment in the pla | vedclass

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(C) Let the given scalene triangle $T$ have vertices $P, Q, R$. We want to find the number of points $A$ such that $	riangle ABC \sim 	riangle PQR$

Vedclass · Accepted Answer

$12$

Vedclass · Answer

(C) Let the given scalene triangle $T$ have vertices $P, Q, R$. We want to find the number of points $A$ such that $	riangle ABC \sim 	riangle PQR$ with the vertices corresponding in the given order.For a fixed line segment $BC$ and a fixed triangle $T$ (with sides $p, q, r$),the similarity $	riangle ABC \sim 	riangle PQR$ implies that the ratio of sides $AB/PQ = BC/QR = AC/PR$ is fixed.For each ordered correspondence of the vertices of $	riangle ABC$ to the vertices of $	riangle PQR$,there are $2$ possible locations for point $A$ (one on each side of the line $BC$).Since there are $3! = 6$ possible permutations of the vertices $P, Q, R$ to match the vertices $A, B, C$ (or more precisely,$6$ ways to map the sides of $T$ to the segment $BC$),and for each mapping,there are $2$ possible positions for $A$ (one on either side of $BC$),the total number of points $A$ is $6 	imes 2 = 12$.

Suppose $BC$ is a given line segment in the plane and $T$ is a scalene triangle. The number of points $A$ in the plane such that the triangle with vertices $A, B, C$ (in that order) is similar to triangle $T$ is

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