Question

what is the relationship between the two triangles shown on the coordinate plane? select two.

△pqr and △stu are not similar but they are congruent because they have the same slope of 1/2.
rq/qp = ut/ts = 1/2
△pqr and △stu are similar because they have the same slope of 1/2.
△pqr and △stu are neither similar nor congruent because they have different slopes.
qp/rq = ts/ut = 1/2
rq/qp = ut/ts = 2
qp/rq = ts/ut = 2
△pqr and △stu are not similar but they are congruent because they have the same slope of 2.
△pqr and △stu are similar because they have the same slope of 2.

Explanation:

Step1: Calculate the lengths of sides

For \(\triangle PQR\), \(QP = 2\), \(RQ= 1\). For \(\triangle STU\), \(TS = 4\), \(UT = 2\).

Step2: Calculate the ratios

\(\frac{RQ}{QP}=\frac{1}{2}\), \(\frac{UT}{TS}=\frac{2}{4}=\frac{1}{2}\). Also, since the slope of the line passing through the triangles is \(\frac{\text{rise}}{\text{run}}\). For both triangles, if we consider the vertical and horizontal segments (e.g., for \(\triangle PQR\) vertical segment \(RQ = 1\), horizontal segment \(QP=2\); for \(\triangle STU\) vertical segment \(UT = 2\), horizontal segment \(TS = 4\)), the slope \(m=\frac{1}{2}\). Similar triangles have equal slopes (corresponding angles equal as they are right - angled and the ratio of sides \(\frac{RQ}{QP}=\frac{UT}{TS}\)).

Answer:

\(\frac{RQ}{QP}=\frac{UT}{TS}=\frac{1}{2}\), \(\triangle PQR\) and \(\triangle STU\) are similar because they have the same slope of \(\frac{1}{2}\)