Constructions (Circles)

Vardaan Learning Institute • ICSE Class 10 Mathematics • Chapter 19

1. Key Instruments and Examination Rules

ICSE Rule: All construction lines and arcs must be shown clearly. Do NOT erase construction arcs. Use a sharp pencil and ruler.

2. Construction of Tangents to a Circle

Type 1: Tangent from an External Point $P$

  1. Draw the given circle $C(O, r)$ and mark the external point $P$ at distance $OP = d$.
  2. Join $OP$ and construct its perpendicular bisector to find the midpoint $M$ of $OP$.
  3. With $M$ as center and radius $MO = MP$, draw an auxiliary circle (or arcs) intersecting the given circle at points $A$ and $B$.
  4. Join $PA$ and $PB$. $PA$ and $PB$ are the required tangents from external point $P$.

Type 2: Tangent at a Point $P$ on the Circle (Without Using Center)

  1. Draw a chord $AB$ through the point $P$.
  2. Take any point $C$ on the major arc $AB$ and join $AC$ and $BC$ to form $\triangle ABC$.
  3. At point $P$, construct $\angle APT$ equal to $\angle ACB$ in the alternate segment.
  4. The line $TPT'$ is the required tangent at point $P$.

3. Circumcircle and Incircle of a Triangle

Circumcircle of $\triangle ABC$

Definition: A circle passing through all three vertices of $\triangle ABC$.

Steps:

  1. Draw the perpendicular bisectors of any two sides (e.g., $AB$ and $BC$).
  2. Let the perpendicular bisectors intersect at point $O$ (Circumcenter).
  3. With $O$ as center and radius $OA = OB = OC$, draw the circle.

Incircle of $\triangle ABC$

Definition: A circle touching all three sides of $\triangle ABC$ internally.

Steps:

  1. Draw the internal angle bisectors of any two angles (e.g., $\angle A$ and $\angle B$).
  2. Let the angle bisectors intersect at point $I$ (Incenter).
  3. From $I$, drop a perpendicular $ID$ onto side $AB$.
  4. With $I$ as center and radius $r = ID$, draw the circle.

4. Construction of Regular Hexagon inside/outside a Circle

To construct a regular hexagon of side $r$ inscribed in a circle of radius $r$:

  1. Draw a circle of radius $r$.
  2. With radius $r$ unchanged on the compass, start at any point $A$ on the circle and step off arcs along the circumference to find vertices $B, C, D, E, F$.
  3. Join $AB, BC, CD, DE, EF, FA$ to form the regular hexagon.

Exam Practice Questions (ICSE PYQs)

PYQ 2023

BOARD Construct a $\triangle ABC$ in which $AB = 6\text{ cm}$, $BC = 7\text{ cm}$, and $\angle ABC = 60^\circ$. Construct the circumcircle of $\triangle ABC$ and measure its radius.

PYQ 2020

BOARD Draw a circle of radius $3.5\text{ cm}$. From a point $P$, $7\text{ cm}$ away from its center, construct two tangents to the circle. Measure the length of each tangent.