L1 · Constructing perpendicular and parallel line segments

Learning goal

“I can state and order the compass-and-straightedge steps for constructing a perpendicular at a point on a line and a parallel through a point, and I know the properties that make the results exact: perpendicular means 90°, and parallels stay the same distance apart everywhere.”

Perpendicular — meeting at exactly 90° (a right angle), like the corner of a page.

Parallel — always the same distance apart and never meeting, however far extended, like railway tracks.

Straightedge — an unmarked ruler used in constructions only for drawing straight lines - lengths are never measured off it; the compass carries all lengths.

Arc — part of a circle drawn by the compass; construction arcs mark off equal distances from a point.

Worked example

Construct a line perpendicular to line l at point P on l. Step 1: Plant the compass point on P. Draw arcs that cross l on BOTH sides of P. Call the crossing points A and B. (PA = PB, because both are radii of the same arc setting.) Step 2: Open the compass wider. From A, draw an arc above the line. WITHOUT changing the opening, from B draw an arc that crosses it. Call the crossing point Q. Step 3: Draw the straight line through P and Q (straightedge). That line is perpendicular to l at P - it meets l at exactly 90°. Say it plainly: the equal arcs force Q to sit exactly over P. Equal distances are what make the right angle exact - it is built, not eyeballed.

Check yourself — explain it

In Step 2, the arcs from A and from B must use the same compass opening because...

Paired problem — example still visible

In the worked example, what is drawn immediately after the arcs from A and B cross at Q? -> The straight line through P and Q

Faded completion — last step is yours

"Perpendicular at P: 1) arcs from P cross the line at A and B; 2) with a wider, unchanged opening, arcs from A and B cross at ___; 3) draw the line through ___ and ___." -> Q; P and Q

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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