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Formulas: geometry, grade 11 (4 units)

The equation of a circle

  • The canonical equation – the centre at the origin, the radius R\displaystyle R
    x2+y2=R2\displaystyle x^{2}+y^{2}=R^{2}
  • The general equation – the centre (a,b)\displaystyle (a,b), the radius R\displaystyle R
    (x−a)2+(y−b)2=R2\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}
  • The distance between two points
    d\displaystyle d =\displaystyle = (x2−x1)2+(y2−y1)2\displaystyle \sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}
  • The midpoint of a segment
    xM=x1+x22, yM=y1+y22\displaystyle x_M=\frac{x_1+x_2}{2},\ y_M=\frac{y_1+y_2}{2}
  • Perpendicular lines – m1\displaystyle m_1, m2\displaystyle m_2 their slopes
    m1⋅m2\displaystyle m_1\cdot m_2 =\displaystyle = −1\displaystyle -1

Angles and chords in a circle

  • An inscribed angle – half the central angle on the same arc
    ∠ACB\displaystyle \angle ACB =\displaystyle = 12∠AOB\displaystyle \frac{1}{2}\angle AOB
  • A quadrilateral inscribed in a circle – a pair of opposite angles
    ∠A+∠C\displaystyle \angle A+\angle C =\displaystyle = 180∘\displaystyle 180^{\circ}
  • A chord AB\displaystyle AB – d\displaystyle d its distance from the centre
    (AB2)2+d2\displaystyle \left(\frac{AB}{2}\right)^{2}+d^{2} =\displaystyle = R2\displaystyle R^{2}

Tangents

  • Two tangents from a point P\displaystyle P – A\displaystyle A, B\displaystyle B the points of contact
    PA=PB\displaystyle PA=PB
  • The tangent segment PA\displaystyle PA – O\displaystyle O the centre
    PA2+R2\displaystyle PA^{2}+R^{2} =\displaystyle = PO2\displaystyle PO^{2}

Trigonometry, circumference and area

  • The law of sines – R\displaystyle R the radius of the circle through the triangle’s vertices
    asin⁡α=bsin⁡β=csin⁡γ=2R\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R
  • The area of a triangle – γ\displaystyle \gamma the angle between the sides a\displaystyle a and b\displaystyle b
    A\displaystyle A =\displaystyle = 12absin⁡γ\displaystyle \frac{1}{2}ab\sin\gamma
  • The area of a parallelogram – a\displaystyle a, b\displaystyle b adjacent sides, α\displaystyle \alpha the angle between them
    A\displaystyle A =\displaystyle = absin⁡α\displaystyle ab\sin\alpha
  • The sine of an obtuse angle
    sin⁡(180∘−α)\displaystyle \sin(180^{\circ}-\alpha) =\displaystyle = sin⁡α\displaystyle \sin\alpha
  • The circumference
    C\displaystyle C =\displaystyle = 2πR\displaystyle 2\pi R
  • The area of a disc
    A\displaystyle A =\displaystyle = πR2\displaystyle \pi R^{2}