Question

△xyz has vertices at x(1, -2), y(9, 2), and z(5, -10). is △xyz an isosceles right triangle? justify your answer. yes, (overline{xy}) and (overline{yz}) are congruent and perpendicular. yes, (overline{xy}) and (overline{xz}) are congruent and perpendicular. no, none of the sides are congruent. no, none of the sides are perpendicular.

Explanation:

Step1: Calculate the lengths of the sides using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For \(XY\) with \(X(1,-2)\) and \(Y(9,2)\):
\(XY=\sqrt{(9 - 1)^2+(2+ 2)^2}=\sqrt{64 + 16}=\sqrt{80}=4\sqrt{5}\)
For \(XZ\) with \(X(1,-2)\) and \(Z(5,-10)\):
\(XZ=\sqrt{(5 - 1)^2+(-10 + 2)^2}=\sqrt{16+64}=\sqrt{80}=4\sqrt{5}\)
For \(YZ\) with \(Y(9,2)\) and \(Z(5,-10)\):
\(YZ=\sqrt{(5 - 9)^2+(-10 - 2)^2}=\sqrt{16 + 144}=\sqrt{160}=4\sqrt{10}\)

Step2: Check if two sides are congruent

Since \(XY = XZ=4\sqrt{5}\), two sides are congruent.

Step3: Check if the two congruent sides are perpendicular using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\)

Slope of \(XY\): \(m_{XY}=\frac{2+2}{9 - 1}=\frac{4}{8}=\frac{1}{2}\)
Slope of \(XZ\): \(m_{XZ}=\frac{-10 + 2}{5 - 1}=\frac{-8}{4}=-2\)
Since \(m_{XY}\times m_{XZ}=\frac{1}{2}\times(-2)=-1\), \(XY\) and \(XZ\) are perpendicular.

Answer:

Yes, \(\overline{XY}\) and \(\overline{XZ}\) are congruent and perpendicular.