Grade 10 power mechanics Related Drawing – Tangency Notes

1. (a) Identify and outline the sub-sub-strands:
   • Methods of constructing tangents
   • Construction of tangent to a circle from a point
   • Construction of external tangents to two circles
   • Construction of internal tangents to two circles
   • Importance of tangency in drawing
2. (b) Describe the methods of constructing tangents in drawing.
3. (c) Construct a tangent to a circle from a point (compass & straightedge).
4. (d) Construct external tangents to two circles (non‑intersecting circles).
5. (e) Construct internal tangents to two circles (non‑intersecting circles).
6. (f) Appreciate the importance of tangency in drawing and practical power‑mechanics work.

A line is tangent to a circle if it touches the circle at exactly one point. At the point of contact the tangent is perpendicular to the radius drawn to that point. Tangency is used frequently in machine drawing and power mechanics when parts touch without cutting into each other (bearings, pulleys, cams, gearing clearances).

• Perpendicular-radius method: If the point of contact is known, draw the radius to that point then draw a line perpendicular to the radius; this line is the tangent.
• Tangent from an external point (Thales method): Use the circle with diameter joining the external point and the circle centre — intersections give tangent points (because angle in semicircle is 90°).
• Homothety / auxiliary circle method (for two circles): Reduce or enlarge one circle by adding or subtracting the other radius (form an auxiliary circle) so both become equal; construct tangents to the auxiliary circle and translate them to the original circles to get external or internal tangents.
• Direct geometric construction: Use straightedge and compass steps (detailed below) to find tangent points and draw the tangent lines precisely.

1. Given circle with centre O and external point P.
2. Join O and P with a straight line (draw OP).
3. Draw the circle having OP as diameter (that is, centre at midpoint M of OP, radius = 1/2 OP).
4. The circle with diameter OP meets the given circle at one or two points (T1 and T2). These are the points of tangency.
5. Join P to each tangent point T1 and T2. Lines PT1 and PT2 are the tangents — they touch the circle once and make a right angle with OT at T.

Use when two circles do not overlap. External tangents touch both circles on the outside and do not cross the line joining centres.

1. Given two circles with centres O1 and O2 and radii r1 and r2 (assume r1 ≥ r2).
2. Construct an auxiliary circle centred at O1 with radius (r1 − r2). If r1 = r2 the auxiliary circle has radius 0 (a point).
3. From O2 draw tangents to this auxiliary circle (use straightedge & compass to draw tangents from a point to a circle — same method as (c)). The tangent points define lines joining O2 to each tangent point.
4. Construct the lines through these tangent points but offset them outward by distance r2 parallel to themselves (equivalently, draw the corresponding lines that are tangent to the original circles). Practically, once you have the tangent line to the auxiliary circle, translate it outward by r2 (draw a line parallel at distance r2) so it touches both original circles — those are the external tangents.
5. Verify: each tangent line touches each circle at a single point and is perpendicular to the radii at those points.

Internal tangents cross the line joining centres between the circles and touch each circle on opposite sides.

1. Given two non‑overlapping circles with centres O1 and O2 and radii r1 and r2.
2. Construct an auxiliary circle centred at O1 with radius (r1 + r2) — add the radii.
3. From O2 draw tangents to this auxiliary circle. Those tangent directions correspond to internal tangents of the original pair.
4. Translate back by reducing each tangent line by r2 on the side of O2 (or construct parallel lines at distance r2) to get the actual tangent lines that touch the original circles internally.
5. Check that each resultant line meets each circle exactly once and the radii at the touch points are perpendicular to the line.

• Accurate representation of contact between parts (pulleys, belts, shafts, bearings) — tangency ensures correct clearance and alignment.
• Used in designing cams and followers where smooth contact is needed.
• Essential in drafting mating parts so that movement and load transfer are correctly shown.
• Precision drawing skills (compass & straightedge) build good workshop practice and help reduce assembly errors.

• Teacher demonstration: show the tangent‑from‑point construction on the board with compass and straightedge using a real circle template.
• Student practical: each learner repeats the constructions on squared paper — tangent from a point, then external and internal tangents to two circles of given sizes.
• Group activity: give pairs two circles drawn on card; students cut out and use pins and thread to explore tangency lines physically (helps visualise contact).
• Application problem: draw pulleys (two different diameters) and construct belt lines using external tangents — relate to power transmission in machines.
• Assessment: short test where learners must produce accurate constructions, label centres and tangent points, and explain checks (radii perpendicular to tangent).

• Always mark centres and radii clearly before constructing tangents.
• Use light construction lines first (pencil) and darken final tangent lines.
• Verify tangency by checking the radius at the contact point is perpendicular to the tangent line.
• If two circles overlap, external/internal tangent constructions above do not apply — first check positions of circles.