| --- |
| 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). |
