Question

you do: check for understanding a triangular park is mapped using coordinates: p(1, 1), q(9, 1), and r(5, 7). city planners want to verify the triangle is isosceles before placing benches symmetrically. is the triangle isosceles? a. yes b. no

Explanation:

Step1: Recall distance formula

The distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Calculate \(PQ\)

For \(P(1,1)\) and \(Q(9,1)\), \(x_1 = 1,y_1 = 1,x_2 = 9,y_2 = 1\).
\(PQ=\sqrt{(9 - 1)^2+(1 - 1)^2}=\sqrt{8^2+0^2}=\sqrt{64}=8\).

Step3: Calculate \(QR\)

For \(Q(9,1)\) and \(R(5,7)\), \(x_1 = 9,y_1 = 1,x_2 = 5,y_2 = 7\).
\(QR=\sqrt{(5 - 9)^2+(7 - 1)^2}=\sqrt{(- 4)^2+6^2}=\sqrt{16 + 36}=\sqrt{52}=2\sqrt{13}\).

Step4: Calculate \(PR\)

For \(P(1,1)\) and \(R(5,7)\), \(x_1 = 1,y_1 = 1,x_2 = 5,y_2 = 7\).
\(PR=\sqrt{(5 - 1)^2+(7 - 1)^2}=\sqrt{4^2+6^2}=\sqrt{16 + 36}=\sqrt{52}=2\sqrt{13}\).

Step5: Compare sides

Since \(QR = PR=2\sqrt{13}\), two sides are equal. So the triangle is isosceles.

Answer:

A. Yes