Question

the graph shows triangle pqr with vertices p(0,2), q(6,4), and r(4,0) and line segment su with endpoints s(4,8) and u(12,4).
at what coordinates would vertex t be placed to create triangle stu, a triangle similar to triangle pqr?
○a. (18,14)
○b. (16,12)
○c. (16,10)
○d. (14,12)

Explanation:

Step1: Calculate the length of sides in triangle PQR

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(PQ\): \(x_1 = 0,y_1 = 2,x_2=6,y_2 = 4\), \(PQ=\sqrt{(6 - 0)^2+(4 - 2)^2}=\sqrt{36 + 4}=\sqrt{40}=2\sqrt{10}\).
For \(QR\): \(x_1 = 6,y_1 = 4,x_2=4,y_2 = 0\), \(QR=\sqrt{(4 - 6)^2+(0 - 4)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\).
For \(PR\): \(x_1 = 0,y_1 = 2,x_2=4,y_2 = 0\), \(PR=\sqrt{(4 - 0)^2+(0 - 2)^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}\).
For \(SU\): \(x_1 = 4,y_1 = 8,x_2=12,y_2 = 4\), \(SU=\sqrt{(12 - 4)^2+(4 - 8)^2}=\sqrt{64 + 16}=\sqrt{80}=4\sqrt{5}\).
Since \(\triangle STU\sim\triangle PQR\), the ratio of similarity \(k=\frac{SU}{QR}=\frac{4\sqrt{5}}{2\sqrt{5}} = 2\).

Step2: Analyze the coordinates

If we assume the correspondence \(P\to S\), \(Q\to T\), \(R\to U\).
The translation and scaling:
The vector from \(P(0,2)\) to \(Q(6,4)\) is \((6 - 0,4 - 2)=(6,2)\).
After scaling by \(k = 2\), the vector for \(S(4,8)\) to \(T\) is \((12,4)\).
\(x\) - coordinate of \(T\): \(4+12=16\), \(y\) - coordinate of \(T\): \(8 + 4=12\).

Answer:

B. \((16,12)\)