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source: ncert
topic: geometry
class_level: class_10
chapter: ch11
difficulty: intermediate
---
Class 10 | Ch11: Constructions
CONSTRUCTION 1: DIVIDE LINE SEGMENT IN GIVEN RATIO (m:n)
To divide AB in ratio 3:2:
Step 1: Draw ray AX at acute angle to AB.
Step 2: Mark 3+2=5 equal arcs on AX: A1,A2,A3,A4,A5.
Step 3: Join A5 to B.
Step 4: Draw line through A3 parallel to A5B (using corresponding angles).
Step 5: This line meets AB at P. AP:PB=3:2.
CONSTRUCTION 2: CONSTRUCT SIMILAR TRIANGLE (scale factor m/n)
To construct triangle similar to ABC with scale factor 3/4:
Case 1 (scale factor < 1, smaller triangle):
Step 1: Draw ray BX at acute angle to BC.
Step 2: Mark 4 equal points on BX: B1,B2,B3,B4.
Step 3: Join B4 to C. Draw B3C' parallel to B4C.
Step 4: Draw C'A' parallel to CA. Triangle A'BC' is required (3/4 times original).
Case 2 (scale factor > 1, larger triangle):
Mark more points (m points), join nth point to vertex.
CONSTRUCTION 3: TANGENTS FROM EXTERNAL POINT
To draw tangents from point P to circle with centre O:
Step 1: Find midpoint M of OP.
Step 2: Draw circle with centre M and radius MP(=MO).
Step 3: This circle intersects original circle at A and B.
Step 4: PA and PB are the required tangents.
Why: Angle OAP=90° (angle in semicircle on diameter OP).
CONSTRUCTION 4: TANGENT AT POINT ON CIRCLE
Draw radius to point. Tangent is perpendicular to radius at that point.
SOLVED EXAMPLES:
Example 1: Divide 7cm segment in ratio 3:2.
Draw 5 equal arcs. Join 5th to B. Through 3rd draw parallel. Done.
Example 2: Tangents from point 10cm from centre, radius 4cm.
Tangent length=sqrt(100-16)=sqrt(84)=2sqrt(21)≈9.17cm.
COMMON MISTAKES:
Scale factor 3/4: mark 4 points (denominator), use 3rd (numerator) for the construction.
Tangent construction: the second circle must have DIAMETER = OP (radius = OP/2). |