IB Maths AI · Topic 3 · 32 cards

Geometry and trigonometry flashcards

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How do you find the distance between two points in 3D? Find it for \((1,2,3)\) and \((4,6,15)\).
\(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}=\sqrt{9+16+144}=13\).
What is the midpoint formula for \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\)?
\(M=\left(\dfrac{x_1+x_2}2,\dfrac{y_1+y_2}2,\dfrac{z_1+z_2}2\right)\): average each coordinate.
What are the volume formulas for a pyramid, a cone and a sphere? Find the volume of a sphere of radius 3.
Pyramid \(\tfrac13Ah\), cone \(\tfrac13\pi r^2h\), sphere \(\tfrac43\pi r^3\). For \(r=3\): \(36\pi\approx113.1\).
What are the curved surface area formulas for a cylinder, a cone and the surface area of a sphere?
Cylinder \(2\pi rh\); cone \(\pi rl\) (\(l\) = slant height); sphere \(4\pi r^2\). A closed solid also needs its flat faces added.
How do you find the curved surface area of a cone with radius 3 and height 4?
Slant height \(l=\sqrt{3^2+4^2}=5\), so curved area \(=\pi rl=15\pi\approx47.1\).
What do SOH CAH TOA say, and how do you find a side? Find the side opposite \(35^\circ\) if the hypotenuse is 12.
\(\sin\theta=\dfrac{opp}{hyp}\), \(\cos\theta=\dfrac{adj}{hyp}\), \(\tan\theta=\dfrac{opp}{adj}\). Opposite \(=12\sin35^\circ\approx6.88\).
How are gradient and angle of inclination linked? Find the angle for gradient 0.5.
\(m=\tan\theta\), so \(\theta=\tan^{-1}m\). For \(m=0.5\), \(\theta\approx26.6^\circ\).
What is the sine rule, and when do you use it?
\(\dfrac a{\sin A}=\dfrac b{\sin B}=\dfrac c{\sin C}\). Use it when you know a side and its opposite angle plus one more angle or side, in a non-right-angled triangle.
What is the ambiguous case of the sine rule?
Given two sides and a non-included angle (SSA), there can be 0, 1 or 2 triangles. Two arise when the angle is acute and the side opposite it is shorter than the other given side but longer than \(b\sin A\).
What is the cosine rule, and when do you use it? Find \(c\) when \(a=5\), \(b=7\), \(C=60^\circ\).
\(c^2=a^2+b^2-2ab\cos C\). Use it for SAS (to find the third side) or SSS (to find an angle). Here \(c^2=25+49-35=39\), so \(c\approx6.24\).
What is the area formula for a triangle with two sides and the included angle? Use it for 5, 7 and \(60^\circ\).
\(A=\tfrac12ab\sin C=\tfrac12(5)(7)\sin60^\circ\approx15.2\).
How do you choose between the sine rule and the cosine rule?
Sine rule: you have a matching side-angle pair. Cosine rule: you have SAS or SSS (no matching pair). Check your GDC is in degree mode before calculating.
What are angles of elevation and depression?
Both are measured from the horizontal: elevation is looking up from the horizontal to an object; depression is looking down. Alternate angles make them equal on parallel horizontals.
How are bearings measured? If B is on a bearing of 060° from A, what is the bearing of A from B?
Clockwise from north, always written with three digits. Back-bearing: \(60+180=240\), so A from B is 240°.
How do you find the angle between a line and a plane in a 3D solid?
Drop a perpendicular from the line's top end to the plane, forming a right-angled triangle with the line's projection on the plane. In a 3×4×12 box the diagonal makes \(\tan^{-1}\frac{12}{5}\approx67.4^\circ\) with the base.
How do you find the arc length and sector area when the angle is in degrees?
\(\text{arc}=\dfrac\theta{360}\times2\pi r\) and \(\text{area}=\dfrac\theta{360}\times\pi r^2\). It is the fraction of the full circle.
What is a Voronoi diagram, and what do its edges and vertices represent?
A division of the plane into cells, one per site, where each cell contains the points closer to that site than any other. Edges lie on perpendicular bisectors of two sites; vertices are equidistant from three or more sites.
How do you find the perpendicular bisector of A(0, 0) and B(4, 2)?
Midpoint \((2,1)\); gradient of AB is \(\tfrac12\), so the perpendicular gradient is \(-2\). \(y-1=-2(x-2)\), i.e. \(y=-2x+5\).
How are Voronoi diagrams used to place a new facility as far as possible from existing sites?
The best location is a Voronoi vertex (or boundary point) with the largest distance to its nearest sites, since it is the centre of the largest empty circle among the sites.
What is nearest-neighbour interpolation on a Voronoi diagram?
To estimate a value at a point, use the known value of the site whose cell contains that point (its closest site).
(HL) How do you convert between degrees and radians? Convert \(150^\circ\).
\(\pi\) rad \(=180^\circ\). Multiply degrees by \(\dfrac\pi{180}\): \(150^\circ=\dfrac{5\pi}6\) rad.
(HL) What are the arc length and sector area formulas when \(\theta\) is in radians? Use \(r=6\), \(\theta=0.5\).
\(s=r\theta\) and \(A=\tfrac12r^2\theta\). Here \(s=3\) and \(A=9\).
(HL) How do you find the vector \(\overrightarrow{AB}\)? Find it for \(A(1,2,3)\) and \(B(4,6,8)\).
\(\overrightarrow{AB}=\mathbf b-\mathbf a\) (end minus start). Here \(\begin{pmatrix}3\\4\\5\end{pmatrix}\), with magnitude \(\sqrt{50}\).
(HL) What is the vector equation of a line?
\(\mathbf r=\mathbf a+t\mathbf d\), where \(\mathbf a\) is a position vector of a point on the line, \(\mathbf d\) is a direction vector and \(t\) is a real parameter.
(HL) What is the scalar product and what does it tell you? Test \((1,2,3)\) and \((2,-1,0)\).
\(\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta\), also the sum of products of components. It is 0 for perpendicular vectors. Here \(2-2+0=0\), so they are perpendicular.
(HL) How do you decide whether two lines in 3D intersect?
Equate the components of the two vector equations. Solve two components for the parameters, then test the third. If it holds they meet; if not and the lines are not parallel, they are skew.
(HL) How do you model an object moving with constant velocity, and when do two objects collide?
\(\mathbf r=\mathbf r_0+t\mathbf v\), with speed \(|\mathbf v|\). They collide only if both position vectors are equal at the same value of \(t\); matching paths at different times is not a collision.
(HL) When does a graph have an Eulerian circuit or trail?
Eulerian circuit: connected and every vertex has even degree. Eulerian trail (not closed): exactly two vertices have odd degree, and the trail starts and ends there.
(HL) What does the adjacency matrix of a graph show, and what do its powers show?
Entry \((i,j)\) counts the edges from vertex \(i\) to vertex \(j\). The entries of \(A^n\) count the walks of length \(n\) between each pair of vertices.
(HL) How do Kruskal's and Prim's algorithms find a minimum spanning tree?
Kruskal: repeatedly add the cheapest remaining edge that does not form a cycle. Prim: start at a vertex and keep adding the cheapest edge joining the tree to a new vertex. A tree on \(n\) vertices has \(n-1\) edges.
(HL) What are the Chinese postman and travelling salesman problems, and how do they differ?
Chinese postman: cover every edge at least once and return, repeating the cheapest set of edges that pairs up the odd vertices. Travelling salesman: visit every vertex once and return at minimum total weight.
(HL) What is the matrix for rotating points anticlockwise by \(\theta\) about the origin?
\(\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}\). Multiply it by the position vector. For \(90^\circ\), \((x,y)\) maps to \((-y,x)\).

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