Constructions
This domain tests ruler-and-compass constructions: dividing a segment in a given ratio, constructing a triangle similar to a given triangle with a given scale factor, and drawing tangents to a circle from an external point.
Core rules
- To divide AB in ratio m : n, draw a ray AX making an acute angle with AB, mark m + n equal arcs, join the last point to B, and draw a parallel through the m-th point.
- To build a triangle similar to △ABC with scale factor k, construct an acute angle at a base vertex, mark equal arcs, and use parallels to copy the base lengths.
- To draw tangents from external point P to a circle with centre O, draw the circle with OP as diameter; its intersections with the given circle are the points of contact.
- Construction lines and arcs should be shown; each point must be labelled.
Worked example
Divide a 7 cm segment AB in the ratio 3 : 2. Draw ray AX, mark A₁, A₂, A₃, A₄, A₅ at equal intervals. Join A₅ to B. Through A₃ draw a line parallel to A₅B meeting AB at C. Then AC : CB = 3 : 2, so AC = 4.2 cm and CB = 2.8 cm.
Traps
Marking m arcs instead of m + n; drawing the parallel through the wrong division point; placing the similar triangle on the wrong side and getting a different correspondence; not drawing the perpendicular bisector of OP when locating tangent points; leaving construction arcs unlabeled.