Question

Step 1: Start by drawing points P and Q. Then use a straightedge to draw line PQ passing through points P and Q. Step 2: Draw point R, not on PQ. Step 3: Use a straightedge to draw line PR passing through points P and R. Step 4: Set the width of a compass to about one-third the distance between P and R. Draw a circle with the center at P. Step 5: Label the intersection of ⨀P and PR as point D. It should fall between points P and R. Label the intersection of ⨀P and PQ as point C. It should fall between points P and Q. Step 6: Using the same compass setting from Step 4, draw a circle with the center at R. Label the point of intersection of ⨀R and PR as point S. It should fall between points P and R. Label the other point of intersection as T. Step 7: Set the width of the compass to the distance between D and C. Step 8: Using this compass setting, draw a circle with the center at T. Label the points of intersection of ⨀R with ⨀T as points F and G. Step 9: Use a straightedge to draw either line RG or RF, whichever one appears to be parallel to line PQ.

Answer 1

In this geometric construction, the final step involves drawing line RF parallel to line PQ. This is achieved by using the circles ⨀R and ⨀T. The previous steps establish points P, Q, R, D, C, S, and T through the use of straightedges and compass. Circles with centers at P and R are drawn, and their intersections with the lines PR and PQ are labeled as D, C, S, and T. By setting the compass width to the distance between D and C, a circle is drawn with center T, intersecting ⨀R at points F and G. The final step, step 9, instructs to draw line RF or line RG, with RF being chosen to be parallel to line PQ.

Explanation:

In this geometric construction, the goal is to draw a line parallel to line PQ through a series of carefully orchestrated steps. Initially, points P and Q are established, and a line PQ is drawn using a straightedge. Point R is then introduced off line PQ, and another line PR is drawn through points P and R. Two circles, ⨀P and ⨀R, are subsequently created with their centers at P and R, respectively, by setting the compass width to one-third the distance between P and R.

The intersections of these circles with lines PR and PQ are labeled as D, C, S, and T. The compass width is adjusted to match the distance between D and C, and a new circle is drawn with center T, intersecting ⨀R at points F and G. Finally, in step 9, a line RF is drawn using the same compass width, ensuring it is parallel to the original line PQ.

This construction method showcases fundamental geometric principles involving circles, intersections, and parallel lines, illustrating the precise use of compass and straightedge techniques to achieve the desired result.