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Formulas: geometry, grade 10 (5 units)

Introduction to analytic geometry

  • The distance between the points (x1,y1)\displaystyle (x_1,y_1) and (x2,y2)\displaystyle (x_2,y_2)
    d\displaystyle d =\displaystyle = (x1−x2)2+(y1−y2)2\displaystyle \sqrt{(x_1-x_2)^{2}+(y_1-y_2)^{2}}
  • The midpoint of the segment
    M(x1+x22, y1+y22)\displaystyle M\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)
  • The slope of a line through two points
    m\displaystyle m =\displaystyle = y2−y1x2−x1\displaystyle \frac{y_2-y_1}{x_2-x_1}x1≠x2\displaystyle x_1\neq x_2
  • The line of slope m\displaystyle m through the point (x1,y1)\displaystyle (x_1,y_1)
    y−y1\displaystyle y-y_1 =\displaystyle = m(x−x1)\displaystyle m(x-x_1)
  • The explicit equation of a line – m\displaystyle m the slope, b\displaystyle b where it crosses the y\displaystyle y-axis
    y=mx+b\displaystyle y=mx+b
  • The general equation of a line
    Ax+By+C=0\displaystyle Ax+By+C=0
  • Parallel lines
    m1\displaystyle m_1 =\displaystyle = m2\displaystyle m_2
  • Perpendicular lines
    m1⋅m2\displaystyle m_1\cdot m_2 =\displaystyle = −1\displaystyle -1
  • A canonical circle – its centre at the origin, R\displaystyle R its radius
    x2+y2=R2\displaystyle x^{2}+y^{2}=R^{2}
  • The circle with centre (a,b)\displaystyle (a,b) and radius R\displaystyle R
    (x−a)2+(y−b)2=R2\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}

Midsegments and the meeting point of the medians

  • A triangle’s midsegment – DE\displaystyle DE is parallel to BC\displaystyle BC
    DE\displaystyle DE =\displaystyle = 12BC\displaystyle \frac{1}{2}BC
  • A trapezoid’s midsegment – a\displaystyle a, b\displaystyle b the bases
    EF\displaystyle EF =\displaystyle = a+b2\displaystyle \frac{a+b}{2}
  • The meeting point of the medians G\displaystyle G divides the median AD\displaystyle AD in the ratio 2:1\displaystyle 2:1
    AG\displaystyle AG =\displaystyle = 2GD\displaystyle 2GD
  • Pythagoras’ theorem and its converse: the triangle is right-angled, the right angle opposite c\displaystyle c
    a2+b2=c2\displaystyle a^{2}+b^{2}=c^{2}

Proportion and similarity

  • A property of a proportion
    ab=cd ⇒ a+bb=c+dd\displaystyle \frac{a}{b}=\frac{c}{d}\ \Rightarrow\ \frac{a+b}{b}=\frac{c+d}{d}
  • Thales’ theorem – DE∥BC\displaystyle DE\parallel BC
    ADDB\displaystyle \frac{AD}{DB} =\displaystyle = AEEC\displaystyle \frac{AE}{EC}
  • The extended Thales theorem – DE∥BC\displaystyle DE\parallel BC
    ADAB=AEAC=DEBC\displaystyle \frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}
  • The angle bisector theorem – AD\displaystyle AD bisects the angle A\displaystyle A
    BDDC\displaystyle \frac{BD}{DC} =\displaystyle = ABAC\displaystyle \frac{AB}{AC}
  • The similarity ratio – corresponding sides
    a1a2=b1b2=c1c2=k\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=k
  • Heights, medians, perimeters, the radii of the circumscribed and of the inscribed circles
    h1h2=m1m2=P1P2=R1R2=r1r2=k\displaystyle \frac{h_1}{h_2}=\frac{m_1}{m_2}=\frac{P_1}{P_2}=\frac{R_1}{R_2}=\frac{r_1}{r_2}=k
  • The ratio of the areas
    S1S2\displaystyle \frac{S_1}{S_2} =\displaystyle = k2\displaystyle k^{2}

The circle

  • An inscribed angle is half the central angle on the same arc
    ∠ACB\displaystyle \angle ACB =\displaystyle = 12∠AOB\displaystyle \frac{1}{2}\angle AOB
  • An inscribed angle on the diameter AB\displaystyle AB
    ∠ACB=90∘\displaystyle \angle ACB=90^{\circ}
  • A quadrilateral ABCD\displaystyle ABCD inscribed in a circle – opposite angles
    ∠A+∠C\displaystyle \angle A+\angle C =\displaystyle = 180∘\displaystyle 180^{\circ}
  • The angle between the tangent AT\displaystyle AT and the chord AB\displaystyle AB equals an inscribed angle on AB\displaystyle AB from its other side
    ∠BAT\displaystyle \angle BAT =\displaystyle = ∠ACB\displaystyle \angle ACB
  • Two tangents from the point P\displaystyle P
    PA=PB\displaystyle PA=PB
  • A quadrilateral ABCD\displaystyle ABCD circumscribing a circle
    AB+CD\displaystyle AB+CD =\displaystyle = BC+AD\displaystyle BC+AD
  • The chords AB\displaystyle AB and CD\displaystyle CD meet at the point E\displaystyle E
    AE⋅EB\displaystyle AE\cdot EB =\displaystyle = CE⋅ED\displaystyle CE\cdot ED
  • Two secants from the point P\displaystyle P
    PA⋅PB\displaystyle PA\cdot PB =\displaystyle = PC⋅PD\displaystyle PC\cdot PD
  • A tangent PT\displaystyle PT and a secant PAB\displaystyle PAB from the point P\displaystyle P
    PT2\displaystyle PT^{2} =\displaystyle = PA⋅PB\displaystyle PA\cdot PB