From df7abfddd50926e105ac361c2d39be0f8079f1aa Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Fri, 30 Nov 2018 10:18:51 +1100 Subject: [PATCH] ellpises and hyperbolas for spec --- spec/prelim.md | 27 ++++++++++++++++++++++++--- 1 file changed, 24 insertions(+), 3 deletions(-) diff --git a/spec/prelim.md b/spec/prelim.md index d53e5ef..586fc59 100644 --- a/spec/prelim.md +++ b/spec/prelim.md @@ -22,6 +22,10 @@ $$a^2=b^2 - 2bc \cos A$$ ## Geometry +### Area of a triangle + +$A={1 \over 2} a b \sin C$ + ### Parallel lines If parallel lines are crossed by transversal: @@ -38,7 +42,24 @@ Sum of interior angles of $n$-sided polygon is $(n-2) \times 180^\circ$ ### Circle geometry -- ![](graphics/circle-centre-angles.png) The angle at the centre of a circle is twice the angle at the circumference subtended by the arc -- ![](graphics/semicircle-right-angle.png) the angle in a semicircle is a right angle -- ![](graphics/segment-angles.png) angles in the same segment of a circle are equal +- ![](graphics/circle-centre-angles.png){#id .class width=40%} The angle at the centre of a circle is twice the angle at the circumference subtended by the arc +- ![](graphics/semicircle-right-angle.png){#id .class width=40%} the angle in a semicircle is a right angle +- ![](graphics/segment-angles.png){#id .class width=40%} angles in the same segment of a circle are equal - ![]() + + +## Ellipses and hyperbolas + +#### Ellipses + +$${(x-h)^2 \over a^2}+{(y-k)^2 \over b^2} = 1$$ + +#### Hyperbolas + +$${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$ + +- centre at $(h,k)$ +- asymptotes at $y-k=\pm{b \over a}(x-h)$ + +${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$ and ${(y-k)^2 \over b^2} - {(x-h)^2 \over a^2} = 1$ are **conjugate hyperbolas** + -- 2.43.2