Grade 10 Geometric Constructions

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Step-by-Step Learning

Learn how to perform various geometric constructions using only a compass and straightedge.

Example 1: Constructing a Tangent at a Point on the Circle

Construct a tangent to a circle at a given point P on the circle, with center O.

Step 1: Draw a circle with center O and mark a point P on its circumference.

Step 2: Draw the radius OP.

Step 3: Extend the radius OP to a point Q such that P lies between O and Q.

Step 4: With P as center and any convenient radius, draw arcs intersecting OQ at two points, say R and S.

Step 5: With R and S as centers and a radius greater than RP (or SP), draw arcs intersecting each other at a point, say T.

Step 6: Draw a line passing through P and T. This line is the required tangent to the circle at point P. (This construction essentially constructs a perpendicular to the radius at point P).

Example 2: Constructing Tangents from an External Point

Construct tangents to a circle from an external point P.

Step 1: Draw a circle with center O and an external point P.

Step 2: Join O to P.

Step 3: Draw the perpendicular bisector of OP. Let it intersect OP at M.

Step 4: With M as center and radius MO (or MP), draw a circle. This circle will intersect the given circle at two points, say A and B.

Step 5: Join P to A and P to B. PA and PB are the required tangents.

Example 3: Dividing a Line Segment in a Given Ratio

Divide a line segment AB in the ratio m:n, where m and n are positive integers.

Step 1: Draw a line segment AB of the given length.

Step 2: Draw a ray AX starting from A, making an acute angle with AB.

Step 3: Mark m + n points on ray AX at equal distances. Let these points be A₁, A₂, ..., A(m+n).

Step 4: Join B to A(m+n).

Step 5: Through the point A(m), draw a line parallel to BA(m+n), intersecting AB at point C.

Result: The point C divides the line segment AB in the ratio m:n (AC : CB = m : n).

Example 4: Constructing a Similar Triangle (Scale Factor > 1)

Construct a triangle similar to a given triangle ABC with a scale factor greater than 1.

Step 1: Draw the given triangle ABC.

Step 2: Draw a ray BX starting from B, making an acute angle with BC, on the opposite side of vertex A.

Step 3: Mark points on ray BX. If the scale factor is k (where k = m/n and m > n), mark m points B₁, B₂, ..., B(m) at equal distances.

Step 4: Join C to B(n) (the point corresponding to the denominator of the scale factor).

Step 5: Through B(m), draw a line parallel to CB(n), intersecting the extended line segment BC at C'.

Step 6: Through C', draw a line parallel to AC, intersecting the extended line segment BA at A'.

Result: Triangle A'BC' is the required triangle similar to triangle ABC with the given scale factor.

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Ask about a specific construction step or concept.

Example questions: "How to bisect an angle?", "What is the first step to construct a tangent from an external point?", "Explain dividing a segment in ratio 3:2".

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Grade 10 Geometric Constructions

Step-by-Step Learning

Example 1: Constructing a Tangent at a Point on the Circle

Example 2: Constructing Tangents from an External Point

Example 3: Dividing a Line Segment in a Given Ratio

Example 4: Constructing a Similar Triangle (Scale Factor > 1)

Practice Mode

Related Concepts

Compass

Straightedge

Perpendicular Bisector

Angle Bisector

Parallel Lines

Perpendicular Lines

Locus

Auxiliary Construction