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Geometry - Geometrical Constructions and Loci

Grade 8IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Locus: A set of points that satisfy a specific rule or set of conditions.

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Perpendicular Bisector: The locus of points equidistant from two fixed points (A and B). It cuts the line segment AB at 90∘90^\circ exactly in the middle.

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Angle Bisector: The locus of points equidistant from two intersecting lines. It divides an angle into two equal parts.

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Locus from a Point: The set of points at a fixed distance 'r' from a central point, forming a circle with radius 'r'.

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Locus from a Line: The set of points at a fixed distance from a line, forming two parallel lines on either side of the original line with semi-circular ends.

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Region Shading: Using inequalities (e.g., 'less than 5cm from X') to shade areas that satisfy multiple loci conditions simultaneously.

📐Formulae

Equidistant \ from \ points \ A \ and \ B: PA = PB

Equidistant from lines L1 and L2:dist(P,L1)=dist(P,L2)Equidistant \ from \ lines \ L_1 \ and \ L_2: dist(P, L_1) = dist(P, L_2)

Distance from point O:x2+y2=r2 (Coordinate geometry representation)Distance \ from \ point \ O: x^2 + y^2 = r^2 \text{ (Coordinate geometry representation)}

Interior angles of a triangle:∠A+∠B+∠C=180∘Interior \ angles \ of \ a \ triangle: \angle A + \angle B + \angle C = 180^\circ

💡Examples

Problem 1:

Construct the locus of points that are equidistant from two points, P and Q, which are 6 cm apart.

Solution:

  1. Draw a line segment PQ of 6 cm. 2. Set the compass width to more than half of PQ (e.g., 4 cm). 3. Draw arcs above and below the line from point P. 4. Keeping the same compass width, draw arcs from point Q. 5. Draw a straight line through the two points where the arcs intersect.

Explanation:

This straight line is the perpendicular bisector. Every point on this line is the same distance from P as it is from Q.

Problem 2:

Draw the locus of points that are exactly 3 cm away from a fixed point C.

Solution:

  1. Mark point C on the paper. 2. Set the compass to a radius of 3 cm using a ruler. 3. Place the compass point on C and draw a full circle.

Explanation:

The locus of points at a fixed distance from a single point is always a circle with that fixed distance as the radius.

Problem 3:

A dog is tied to a 5m leash attached to a 10m long straight fence. Describe the locus of the area the dog can reach.

Solution:

The locus consists of a rectangle (parallel to the fence) and two semi-circles at the ends of the fence.

Explanation:

Since the dog is constrained by a leash (fixed distance) and a line (the fence), the boundary is 5m away from the line. At the corners/ends of the fence, the dog moves in a circular path, creating semi-circular boundaries.