Grade 7 Mathematics GEOMETRY – Geometrical constructions Notes

Mathematics — GEOMETRY

Subtopic: Geometrical constructions (Age ~12, Kenyan)

In constructions you use only a ruler (straightedge) and a compass to draw exact shapes. Below are simple, important constructions with easy steps and small diagrams. Try them with real ruler & compass.

1. Construct an equilateral triangle on a given segment AB

Given AB, place compass point on A, draw circle with radius AB. With same radius and centre B draw another circle. The circles meet at point C. Join A–C and B–C. Triangle ABC is equilateral (all sides = AB).

Why it works

Any point where the two circles meet is the same distance from A and B (radius AB). So AC = AB and BC = AB — all three sides equal (60° angles).

2. Perpendicular bisector of a line segment AB

This line cuts AB into two equal parts and makes a right angle with it.

1. Place compass at A with radius > half AB; draw arc above and below the line.
2. Without changing compass, draw arcs from B so the two pairs of arcs intersect.
3. Join the intersection points. This line is the perpendicular bisector of AB and meets AB at the midpoint M.

Property: Any point on the perpendicular bisector is equidistant from A and B.

3. Angle bisector (bisect angle ∠BAC)

The angle bisector divides the angle into two equal angles.

1. Place compass at A; draw an arc that cuts both sides AB and AC at points P and Q.
2. With the same compass width, draw arcs from P and Q so they cross at point R.
3. Join A to R. AR is the angle bisector of ∠BAC.

Property: Any point on the angle bisector is equidistant from the two sides of the angle.

4. Perpendicular from a point P (not on line l) to line l

To drop a perpendicular from P to l:

1. With centre P, draw an arc that meets line l at two points X and Y.
2. With centres X and Y and same radius, draw arcs above the line that meet at Z.
3. Join P to Z. The line PZ meets l at a right angle (foot H).

5. Construct a line parallel to line l through a point P

One method uses copying an angle (alternate interior angles):

1. Pick a point A on line l near desired point P.
2. Join P to A. At A make an angle with l equal to angle made by PA and l using compass (copy angle).
3. The line through P at that copied angle is parallel to l (alternate interior angles equal).

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Practice questions

1. Construct the perpendicular bisector of a 6 cm segment. Measure and state the midpoint.
2. Given ∠XAY, bisect it and measure each half angle. (Use protractor to check.)
3. From point P 4 cm above a line, drop a perpendicular to the line. Measure the foot of perpendicular.
4. Construct an equilateral triangle with side 5 cm. Check each side with your ruler.

Answers / hints

Hints:

• Perpendicular bisector: arcs should overlap above and below to get intersection points. Midpoint is where bisector meets segment.
• Angle bisector: use same compass width for arcs from sides so intersections are clear.
• Perpendicular from P: when arcs from X and Y meet, they determine direction of perpendicular.
• Equilateral triangle: the arcs from ends of the 5 cm side must have radius 5 cm.

Tip: Work lightly with your pencil until the construction is finished so you can erase unwanted lines. Practice each step slowly and check with a ruler or protractor.

Prepared for Kenyan learners (age ~12). Try these on paper with a real compass and ruler for best learning.