Question

janet draws triangle pqr with vertices p(1,1), q(-2,4), and r(1,7). she claims that the triangle is an isosceles right triangle. which statement shows that janet is correct? a the slope of pq is 1, which is the reciprocal of the slope of qr, and pq and qr are equal to 3√2. b the slope of pq is -1, which is the negative reciprocal of the slope of qr, and pq and qr are equal to √6. c the slope of pq is 1, which is the reciprocal of the slope of qr, and pq and qr are equal to √6. d the slope of pq is -1, which is the negative reciprocal of the slope of qr, and pq and qr are equal to 3√2.

Explanation:

Step1: Calculate the slope of \(PQ\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(P(1,1)\) and \(Q(-2,4)\), \(m_{PQ}=\frac{4 - 1}{-2 - 1}=\frac{3}{-3}=-1\).

Step2: Calculate the slope of \(QR\)

For points \(Q(-2,4)\) and \(R(1,7)\), \(m_{QR}=\frac{7 - 4}{1-(-2)}=\frac{3}{3}=1\).

Step3: Calculate the length of \(PQ\)

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(P(1,1)\) and \(Q(-2,4)\), \(d_{PQ}=\sqrt{(-2 - 1)^2+(4 - 1)^2}=\sqrt{(-3)^2+3^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\).

Step4: Calculate the length of \(QR\)

For \(Q(-2,4)\) and \(R(1,7)\), \(d_{QR}=\sqrt{(1-(-2))^2+(7 - 4)^2}=\sqrt{(3)^2+3^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}\).

Since the slope of \(PQ=-1\), the slope of \(QR = 1\) (and \(-1\) is the negative reciprocal of \(1\)), and \(PQ = QR=3\sqrt{2}\).

Answer:

D. The slope of \(\overline{PQ}\) is \(-1\), which is the negative reciprocal of the slope of \(\overline{QR}\), and \(PQ\) and \(QR\) are equal to \(3\sqrt{2}\)