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Geometry: loci and constructions — KCSE Mathematics

KCSE Mathematics · 100 practice questions · 3 syllabus objectives · 3 revision lessons

31 easy34 medium35 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Define locus and describe common loci (perpendicular bisector, angle bisector, circle, parallel line)

Construct geometric figures (perpendicular bisector, angle bisector, circumscribed/inscribed circles, regular polygons) using ruler and compass only

Geometry: loci and constructions

Revision Notes

Concise lesson notes for Geometry: loci and constructions, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Understanding Loci in Geometry

In geometry, a locus is a set of points that satisfy a particular condition. Here are some common types of loci:

  • Perpendicular Bisector: This is the locus of points equidistant from the endpoints of a line segment. It is a straight line that divides the segment into two equal parts at a right angle.
  • Angle Bisector: This locus consists of points equidistant from the two sides of an angle. It divides the angle into two equal angles.
  • Circle: The locus of points at a fixed distance (radius) from a central point. All points on the circumference are equidistant from the center.
  • Parallel Line: The locus of points that maintain a constant distance from a given line. These lines never intersect.

Understanding these concepts allows you to solve problems involving distances and relationships between points effectively.

Key points to remember

  • A locus is a set of points meeting specific conditions.
  • Perpendicular bisector is equidistant from segment endpoints.
  • Angle bisector is equidistant from angle sides.
  • A circle's locus is points at a fixed radius from a center.
  • Parallel lines maintain constant distance from a given line.

Worked example

Define the locus of points equidistant from two points A and B.

  • The locus is the perpendicular bisector of the segment AB.

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More lessons in this topic

Lesson 2: Constructing Geometric Figures with Ruler and Compass

Objective: Construct geometric figures (perpendicular bisector, angle bisector, circumscribed/inscribed circles, regular polygons) using ruler and compass only

To construct geometric figures using a ruler and compass, follow these steps:

  1. Perpendicular Bisector:

    • Draw a line segment, say AB.
    • Place the compass on point A, draw arcs above and below the line.
    • Without changing the compass width, repeat from point B to intersect the arcs.
    • Draw a line through the intersection points; this is the perpendicular bisector of AB.

  2. Angle Bisector:

    • Draw an angle, say ∠ABC.
    • With the compass on point B, draw an arc that intersects both rays.
    • Label the intersection points D and E.
    • With the same radius, draw arcs from points D and E; label their intersection F.
    • Draw a line from B to F; this is the angle bisector.

  3. Circumscribed Circle:

    • Construct a triangle ABC.
    • Find the perpendicular bisectors of any two sides, say AB and AC; label their intersection as O.
    • Using O as the center, draw a circle through A, B, and C.

  4. Regular Polygon:

    • For a hexagon, draw a circle, then divide it into 6 equal parts using 60° angles.

Practice these constructions to gain confidence!

  • Use a ruler and compass for all constructions.
  • Ensure accurate measurements for precision.
  • Label all points and lines clearly.
  • Check intersections carefully for accuracy.

Construct a perpendicular bisector of line segment XY. 1. Draw segment XY. 2. Place compass at X, draw arcs above and below. 3. Repeat at Y. 4. Connect intersection points; this is the bisector.

Lesson 3: Understanding Loci and Constructions in Geometry

Objective: Geometry: loci and constructions

In geometry, loci refers to the set of points that satisfy a particular condition. Understanding loci is crucial for solving geometric problems. Here are some common types of loci:

  • Circle: The locus of points equidistant from a fixed point (the center).
  • Perpendicular bisector: The locus of points equidistant from two fixed points.
  • Angle bisector: The locus of points equidistant from the sides of an angle.

Constructions involve drawing geometric figures accurately using a compass and straightedge. Common constructions include:

  • Constructing a perpendicular bisector of a line segment.
  • Constructing an angle bisector.

To solve problems involving loci and constructions, follow these steps:

  1. Identify the conditions given in the problem.
  2. Determine the locus based on those conditions.
  3. Use a compass and straightedge for precise constructions.

For example, if asked to construct the locus of points equidistant from two points A and B, you would:

  • Draw line segment AB.
  • Find the midpoint of AB.
  • Construct a perpendicular line at the midpoint to represent the locus.
  • Loci represent points satisfying specific geometric conditions.
  • Common loci include circles, perpendicular bisectors, and angle bisectors.
  • Constructions use a compass and straightedge for accuracy.
  • Identify conditions before determining the locus.
  • Follow precise steps for accurate constructions.

Construct the locus of points equidistant from points A(2,3) and B(6,7).

  • Find midpoint M of AB: M(4,5).
  • Draw the perpendicular bisector of AB through M.

Sample Questions

Read 3 questions and answers free. Sign up to access all 100 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER4 marks

State the steps to construct the angle bisector of angle ABC using a ruler and compass only. (4 marks)

Answer & marking scheme

Part (a) — 4 marks
Place the compass at point B and draw an arc that intersects both sides of the angle (1 mk)
Label the intersection points as D and E (1 mk)
With the same radius, draw arcs from points D and E (1 mk)
Draw a straight line from B to the intersection of the two arcs (1 mk)
2
easySHORT ANSWER2 marks

State the locus of points that are all equidistant from two fixed points A and B. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
The locus is the perpendicular bisector of the line segment AB (1 mk)
It is a straight line that divides segment AB into two equal parts at right angles (1 mk)
3
easySHORT ANSWER4 marks

Describe the method to construct a circle that is inscribed in triangle XYZ. (4 marks)

Answer & marking scheme

Part (a) — 4 marks
Construct the angle bisectors of two angles in triangle XYZ (1 mk)
Label the intersection of the two angle bisectors as point G (1 mk)
Using a compass, measure the distance from G to one side of the triangle (1 mk)
Draw a circle centred at G with the measured radius to touch all three sides of the triangle (1 mk)
4

Name the steps to construct the angle bisector of angle ABC using only a ruler and compass. (3 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Geometry: loci and constructions" cover?

Geometry: loci and constructions covers Define locus and describe common loci (perpendicular bisector, angle bisector, circle, parallel line); Construct geometric figures (perpendicular bisector, angle bisector, circumscribed/inscribed circles, regular polygons) using ruler and compass only; Geometry: loci and constructions, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Geometry: loci and constructions?

HighMarks has 100 Geometry: loci and constructions practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Geometry: loci and constructions for the KCSE exam?

Start with the revision notes on this page to refresh the core concepts, then work through the practice questions in increasing difficulty. Sign up for HighMarks to get a personalised study plan that adapts to the topics you keep getting wrong, plus mock exams, subject-wide practice, and detailed performance tracking. See pricing.

Why Practise Geometry: loci and constructions?

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