Question

the figure shows a partially - completed set of steps to construct a rhombus pqrs
the next step is to construct side ps of the figure so that side ps is parallel and congruent to side qr. the chart shows the steps each of the four students took to draw side ps
which student used the correct steps to construct rhombus pqrs?
student 1: fix the compass at m, and adjust its width to point l. without changing the width of the compass, move the compass to n and draw a small arc on the big arc. label the point of intersection of the two arcs as t. draw a line segment from p that passes through t. adjust the width of the compass to qr, and draw an arc from point p to intersect line pt at s
student 2: fix the compass at m, and draw an arc that intersects side qp at point t. without changing the width of the compass, move the compass to r and draw an arc. adjust the width of the compass to qr, and draw an arc from point p. draw a line segment from r that passes through t and intersects the second arc at s
student 3: fix the compass at l, and adjust its width to point m. without changing the width of the compass, move the compass to r and draw an arc that intersects qr at point t. draw a line segment from r that passes through t and intersects the arc at s
student 4: fix the compass at l, and adjust its width to point p. without changing the width of the compass, move the compass to r and draw an arc that intersects qr at point t. draw a line segment from r that passes through t and intersects the arc at s

Explanation:

Step1: Analyze Student 1's steps

Student 1 fixes the compass at \(M\), adjusts width to \(L\). Then moves to \(N\) and draws an arc. This is a valid way to construct congruent segments. Then using the compass width of \(QR\) (by adjusting to \(QR\)) and drawing from \(P\) to intersect line \(PT\) at \(S\). This ensures \(PS = QR\) (congruent) and by the construction of parallel lines (using the same - compass - width - based arc - drawing technique for parallelism).

Step2: Analyze Student 2's steps

Student 2 first draws an arc from \(M\) to intersect \(QP\) at \(T\), then from \(R\) with the same width. But the step of adjusting the compass to \(QR\) and drawing from \(P\) is not a standard way to ensure both parallelism and congruence in one - go as per the basic construction principles of parallelograms (a rhombus is a special parallelogram).

Step3: Analyze Student 3's steps

Student 3 fixes the compass at \(L\), adjusts width to \(M\). Moving to \(R\) and drawing an arc. But the process of drawing a line from \(R\) through \(T\) to intersect the arc at \(S\) does not clearly establish the congruence and parallelism relationship with \(QR\) in the way required for a rhombus (a rhombus has all sides equal and opposite sides parallel).

Step4: Analyze Student 4's steps

Student 4 fixes the compass at \(L\), adjusts width to \(P\). Moving to \(R\) and drawing an arc. The step of drawing a line from \(R\) through \(T\) to intersect the arc at \(S\) does not use the length of \(QR\) (a side of the rhombus) in a way that ensures \(PS = QR\) (congruence) and parallelism.

Answer:

d. Student 1