Question

which of the following sets of ordered pairs will produce a triangle with vertices x, y, and z that is congruent to △prt? graph omitted options: z(5, 2), z(0, 2), z(6, 2), z(5, 3)

Explanation:

To solve this, we first analyze \(\triangle PRT\). From the graph, \(R\) and \(T\) seem to have a horizontal distance (let's find coordinates: assume \(R(-4,4)\), \(T(-1,4)\), so horizontal length \(RT = |-1 - (-4)|= 3\). The vertical side (from \(R\) to \(P\)): if \(P(-4,8)\), then vertical length \(PR=|8 - 4| = 4\). For \(\triangle XYZ\) to be congruent, it must have the same side lengths. Let's assume \(X(2,6)\) and \(Y(2,2)\) (from the vertical line). The vertical distance \(XY = |6 - 2| = 4\) (matching \(PR\)). Now, the horizontal distance from \(Y(2,2)\) to \(Z\) should be \(3\) (matching \(RT\)). So \(Z\) should be at \(x = 2 + 3=5\), \(y = 2\). So \(Z(5,2)\).

Step1: Find side lengths of \(\triangle PRT\)

Assume \(R(-4,4)\), \(T(-1,4)\), \(P(-4,8)\).
Horizontal length \(RT = |-1 - (-4)| = 3\).
Vertical length \(PR = |8 - 4| = 4\).

Step2: Analyze \(\triangle XYZ\)

Assume \(X(2,6)\), \(Y(2,2)\). Vertical length \(XY = |6 - 2| = 4\) (matches \(PR\)).

Step3: Determine \(Z\)’s coordinates

To match \(RT = 3\), horizontal distance from \(Y(2,2)\) to \(Z\) must be \(3\).
So \(x\)-coordinate of \(Z\): \(2 + 3 = 5\), \(y\)-coordinate: \(2\) (matches \(Y\)’s \(y\)-coordinate for horizontal side).
Thus, \(Z(5,2)\).

Answer:

Z(5, 2)