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).