Circles: SSC CGL Quantitative Aptitude

Formulas (4)

• Parts of a circle and the equal-chord facts · Longest chord
  $$\text{Diameter} = 2r$$
• Half-chord, distance from the centre and radius · Chord, distance and radius
  $$r^2 = p^2 + \left(\frac{c}{2}\right)^2 \qquad c = 2\sqrt{r^2 - p^2}$$
• A chord of the larger of two concentric circles · Concentric circles
  $$\text{Chord} = 2\sqrt{R^2 - r^2} \qquad R^2 - r^2 = \left(\frac{C}{2}\right)^2 - \left(\frac{c}{2}\right)^2$$
• Two parallel chords and the radius · Two parallel chords
  $$\text{gap} = p_1 - p_2 \text{ (same side)} \qquad \text{gap} = p_1 + p_2 \text{ (opposite sides)}$$

Watch out for (8)

• Calling the radius the longest chord
  → Parts of a circle and the equal-chord facts
• Mixing up a secant and a tangent
  → Parts of a circle and the equal-chord facts
• Putting the whole chord in the right triangle
  → Half-chord, distance from the centre and radius
• Stopping at the half-chord
  → Half-chord, distance from the centre and radius
• Treating the inner radius as half the chord
  → A chord of the larger of two concentric circles
• Taking the point of contact as an end of the chord
  → A chord of the larger of two concentric circles
• Subtracting the distances for chords on opposite sides
  → Two parallel chords and the radius
• Placing the longer chord farther from the centre
  → Two parallel chords and the radius

Formulas (4)

• Angle at the centre is twice the angle at the circumference · Centre and circumference
  $$\angle AOB = 2\,\angle ACB \qquad \text{reflex } \angle AOB = 360^\circ - \angle AOB$$
• Angle in a semicircle and the circle through three points · Diameter and right angle
  $$\angle ACB = 90^\circ \text{ when } AB \text{ is a diameter} \qquad R = \frac{\text{hypotenuse}}{2}$$
• Opposite angles of a cyclic quadrilateral · Cyclic quadrilateral
  $$\angle A + \angle C = 180^\circ \qquad \angle B + \angle D = 180^\circ$$
• A chord's length from the angle it subtends · Chord from the centre angle
  $$c = 2r\sin\frac{\theta}{2}$$

Watch out for (7)

• Doubling instead of halving
  → Angle at the centre is twice the angle at the circumference
• Halving the centre angle for a point on the minor arc
  → Angle at the centre is twice the angle at the circumference
• Taking a shorter side of a right triangle as the diameter
  → Angle in a semicircle and the circle through three points
• Assuming any three points lie on a circle
  → Angle in a semicircle and the circle through three points
• Adding adjacent angles to 180°
  → Opposite angles of a cyclic quadrilateral
• Matching an exterior angle with the wrong interior angle
  → Opposite angles of a cyclic quadrilateral
• Using the angle at the circumference as the centre angle
  → A chord's length from the angle it subtends

Formulas (4)

• Tangent at right angles to the radius, and its length · Length of a tangent
  $$t = \sqrt{d^2 - r^2}$$
• Two tangents from one outside point · Two tangents from P
  $$PA = PB \qquad \angle APB + \angle AOB = 180^\circ \qquad \text{Area}(OAPB) = r \times t$$
• Tangent lengths from the corners of a triangle to its inscribed circle · Tangent from a corner
  $$AF = s - a \qquad s = \frac{a + b + c}{2}$$
• The alternate segment theorem · Tangent and chord
  $$\angle(\text{tangent}, CA) = \angle CBA = \frac{1}{2}\angle COA$$

Watch out for (8)

• Adding the squares when the distance is given
  → Tangent at right angles to the radius, and its length
• Answering with the radius when the diameter is asked
  → Tangent at right angles to the radius, and its length
• Setting the angle at the centre equal to the angle between the tangents
  → Two tangents from one outside point
• Using the whole angle between the tangents in triangle OAP
  → Two tangents from one outside point
• Subtracting a neighbouring side from the semi-perimeter
  → Tangent lengths from the corners of a triangle to its inscribed circle
• Adding the tangent lengths once for the perimeter
  → Tangent lengths from the corners of a triangle to its inscribed circle
• Using the angle in the same segment instead of the alternate one
  → The alternate segment theorem
• Doubling the tangent-chord angle to get the angle at the circumference
  → The alternate segment theorem

Formulas (3)

• Two chords crossing inside a circle · Intersecting chords
  $$PA \times PB = PC \times PD$$
• Secants and a tangent from an outside point · Secants and a tangent from P
  $$PA \times PB = PC \times PD \qquad PT^2 = PA \times PB$$
• Power of a point from its distance to the centre · Power of a point
  $$PA \times PB = d^2 - r^2 \text{ (P outside)} \qquad PA \times PB = r^2 - d^2 \text{ (P inside)}$$

Watch out for (4)

• Multiplying a piece by the whole chord when the chords cross inside
  → Two chords crossing inside a circle
• Using the chord instead of the whole secant
  → Secants and a tangent from an outside point
• Squaring the near length instead of the tangent
  → Secants and a tangent from an outside point
• Using $r^2 - d^2$ for an outside point
  → Power of a point from its distance to the centre

Formulas (3)

• Length of a direct common tangent · Direct common tangent
  $$L = \sqrt{d^2 - (r_1 - r_2)^2}$$
• Length of a transverse common tangent, and the two tangents together · Transverse common tangent
  $$M = \sqrt{d^2 - (r_1 + r_2)^2} \qquad L^2 - M^2 = 4r_1 r_2$$
• The common chord of two intersecting circles · Common chord
  $$h^2 = r_1^2 - x^2 = r_2^2 - (d - x)^2 \qquad \text{Chord} = 2h$$

Reference tables (1)

Watch out for (8)

• Giving intersecting circles a transverse tangent
  → Positions of two circles and the number of common tangents
• Counting three tangents for internal touching
  → Positions of two circles and the number of common tangents
• Using the sum of the radii for a direct tangent
  → Length of a direct common tangent
• Treating the tangent as the hypotenuse
  → Length of a direct common tangent
• Swapping the sum and the difference
  → Length of a transverse common tangent, and the two tangents together
• Finding a transverse tangent for circles that cut
  → Length of a transverse common tangent, and the two tangents together
• Putting the chord's midpoint halfway between unequal circles
  → The common chord of two intersecting circles
• Forgetting to double half the chord
  → The common chord of two intersecting circles

Formulas (2)

• Direct common tangent of two circles touching externally · Touching circles, direct tangent
  $$L = 2\sqrt{r_1 r_2}$$
• Centres, the point of contact, and a circle touching two others · Distance between touching centres
  $$d = r_1 + r_2 \text{ (external)} \qquad d = r_1 - r_2 \text{ (internal)}$$

Watch out for (4)

• Using $2\sqrt{r_1 r_2}$ for circles that do not touch
  → Direct common tangent of two circles touching externally
• Taking the sum of the radii as the tangent
  → Direct common tangent of two circles touching externally
• Adding the radii for circles touching internally
  → Centres, the point of contact, and a circle touching two others
• Using a radius alone for the distance to a third touching circle
  → Centres, the point of contact, and a circle touching two others