Category Archives: Year 11 Specialist Mathematics

Intersecting Secant Theorem

CD is a tangent to the circle.

Prove c^2=a(a+b)

I am going to add two chords to the circle

Chord AD and BD are added

\angle{BDC}=\angle{CAD} (angles in alternate segments are congruent)

\angle{BCD}=\angle{DCA} (shared angle)

\therefore \Delta BDC\cong \Delta{DAC} (AA)

Hence

\frac{DC}{AC}=\frac{BC}{DC} (Corresponding sides in similar triangles)

\frac{c}{a+b}=\frac{a}{c}

\therefore c^2=a(a+b)

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Filed under Circle Theorems, Geometry, Year 11 Specialist Mathematics

Circle Geometry Question 2

One of my Year 11 Specialist students had this question

Triangle ABC touches the given circle at Points P, Q, R and S only. The secant BW touches the circle at V and W.

Diagram not drawn to scale

(a) Determine the lengths of the line segments marked x, y and z, leaving your answers as exact values.

(b) If the length of the line segment QW is 4 units, determine the exact radius of the circle.

(a) We are going to use the Intersecting Secant Theorem – the tangent version

c^2=a\times(a+b)

Hence, we have

    \begin{equation*}30^2=25(25+x+6)\end{equation}

    \begin{equation*}900=25(31+x)\end{equation}

    \begin{equation*}x=5\end{equation}

Then we can use the intersecting chord theorem to find y.

    \begin{equation*}10\times y=6 \times x\end{equation}

    \begin{equation*}10y=30\end{equation}

    \begin{equation*}y=3\end{equation}

Back to the Intersecting Secant Theorem to find z

    \begin{equation*}z^2=4\times 17\end{equation}

    \begin{equation*}z=2\sqrt{17}\end{equation}

(b)


QW is part of a 3-4-5 triangle, therefore \angle{Q}=90^\circ

This is definitely the case of the diagram not being drawn to scale. If \angle{Q}=90^\circ, then the purple line must be the diameter.

We can use pythagoras to find the length of the diameter

    \begin{equation*}(2r)^2=13^2+4^2\end{equation}

    \begin{equation*}4r^2=185\end{equation}

    \begin{equation*}r=\frac{\sqrt{285}}{2}\end{equation}

The radius of the circle is \frac{\sqrt{285}}{2}

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Filed under Circle Theorems, Geometry, Pythagoras, Year 11 Specialist Mathematics

Circle Geometry Question

In the above diagram O is the centre of the larger circle. A, B,D and E are points on the circumference of the larger circle. A, C, E and 0 are points on the circumference of the smaller circle. Show that \angle{CAB}=\angle{ABC}. AB, AC and BC are straight lines.

AO=OB (radii of the larger circle)

At a line from O to E (it is also a radius of the larger circle)

Let \angle{CAB}=\alpha.

ACEO is a cyclic quadrilateral.

Hence, \angle{CED}=180-\alpha (AECO is a cyclic quadrilateral)

As CB is a straight line \angle{OEB}=180-(180-\alpha)=\alpha.

\Delta OEB is an isosceles triangle.

Therefore, \angle{ABC}=\alpha

Therefore \angle{ABC}=\angle{CAB}

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Filed under Circle Theorems, Finding an angle, Geometry, Year 11 Specialist Mathematics

Derangements

From Wikipedia

In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position.

For example, if we have the set {A, B, C}, there are 6 permutations

ABC, ACB, BAC, BCA, CAB, CBA

But only 2 of them are derangements – BCA and CAB

I did a practical question on this here. In that question I used a tree diagram, but there must be a way to determine the number of derangements given n elements.

We know 3 elements has 2 derangements, what about 4 elements? We know there are 4!=24 permutations

ABCDBACDCABDDABC
ACBDBADCCADBDACB
ACDBBCADCBADDBAC
ABDCBCDACBDADBCA
ADBCBDACCDABDCAB
ADCBBDCACDBADCBA

I have highlighted the derangements. when n=4 there are 9 derangements.

Let’s try to generalise.

  • In how many ways can one element be in its original position?
    \begin{pmatrix}4\\1\end{pmatrix} \times 3!=24
    Choose one element from the 4 to be in its original position, and then arrange the remaining three elements.

    So far, we have D_3=4!-\begin{pmatrix}4\\1\end{pmatrix} \times 3!=0

    Clearly we are counting some arrangements multiple times, for example ABDC has A and B in the correct position, so we need to add all of the arrangements with 2 elements in their original position.
  • In how many ways can two elements be in their original position?
    \begin{pmatrix}4\\2\end{pmatrix} \times 2!=12

    So now we have, D_3=4!-\begin{pmatrix}4\\1\end{pmatrix} \times 3!+\begin{pmatrix}4\\2\end{pmatrix} \times 2!=12

    But once again we have counted some arrangements multiple times, so we need to subtract all of the arrangements with 3 elements in their original position.
  • In how many ways can three elements be in their original position?
    \begin{pmatrix}4\\3\end{pmatrix} \times 1!=4

    So now we have, D_3=4!-\begin{pmatrix}4\\1\end{pmatrix} \times 3!+\begin{pmatrix}4\\2\end{pmatrix} \times 2!-\begin{pmatrix}4\\3\end{pmatrix} \times 1!=8

    We now need to add all of the arrangements with 4 elements in their original position.
  • In how many ways can four elements be in their original position?
    \begin{pmatrix}4\\4\end{pmatrix} \times 0!=1

    So now we have, D_3=4!-\begin{pmatrix}4\\1\end{pmatrix} \times 3!+\begin{pmatrix}4\\2\end{pmatrix} \times 2!-\begin{pmatrix}4\\3\end{pmatrix} \times 1!+\begin{pmatrix}4\\4\end{pmatrix} \times 0!=9

Hence,

D_n=\begin{pmatrix}n\\0\end{pmatrix}\times n!-\begin{pmatrix}n\\1\end{pmatrix}\times (n-1)!+\begin{pmatrix}n\\2\end{pmatrix}\times (n-2)!-... \mp \begin{pmatrix}n\\n\end{pmatrix}\times 0!

Which we can simplify to

D_n=\Sigma_{i=0}^{n}(-1)^n\begin{pmatrix}n\\i\end{pmatrix}\times(n-i)!

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Filed under Counting Techniques, Interesting Mathematics, Year 11 Specialist Mathematics

Counting Techniques

Four teachers decide to swap desks at work. How many ways can this be done if no teacher sits at their previous desk?
Mathematics Specialist Units 1&2 Cambridge

I like this question as it seems easy until you start thinking about it. I think the best approach is a tree diagram.

If we think of the four teachers as A, B, C and D. Then A can no longer sit in A, so the options are B, C and D for the first desk.

For the second desk, If B is in the first desk, then A, C or D could be in the second. If C is in the first desk, then A or D could be in the second (B can’t be in the same desk). If D is in the first desk, then A or C can be in the second desk.

And so on, leaving 9 possibilities

BADC
BCDA
BDAC
CADB
CDAB
CDBA
DABC
DCAB
DCBA

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Filed under Counting Techniques, Tree Diagram, Year 11 Specialist Mathematics

Vectors – Closest Approach (2D)

At 10am, object M travelling with constant velocity (20, 10) km/h is sighted at the point with position vector (-100, 100) km. At 11am object N travelling with constant velocity v=(x,y) km/h is sighted at the point with position vector (-20,-50) km respectively. Use a scalar product method to determine v given that the two objects were closest together at a distance of 40 km at 4pm.

OT Lee Mathematics Specialist Year 11

At 4pm M is at the point with position vector (-100+6\times 20, -100+6\times 10)=(20, -40)

and N is at the point with position vector (-20+5\times x, -50+5\times y)=(-20+5x, -50+5y)

We know the distance between M and N at 4pm is 40 km.

Hence,

    \begin{equation*}40^2=(20-(-20+5x))^2+(-40-(-50+5y))^2\end{equation}

    \begin{equation*}40^2=(40-5x)^2+(10-5y)^2\end{equation}

    \begin{equation*}1600=25x^2-400x+1600+25y^2-100y+100\end{equation}

(1)   \begin{equation*}0=x^2+y^2-16x-4y+4\end{equation*}

In the diagram below, I have found the position vector of N relative to M (_Nr_M) and the velocity of N relative to M (N_v_M)

We know that when _Nr_M\cdot_Nv_M=0 M and N are the closest distance apart.

    \begin{equation*}(-40+5x,-10+5y)\cdot(x-20,y-10)=0\end{equation}

    \begin{equation*}(-40+5x)(x-20)+(-10+5y)(y-10)\end{equation}

    \begin{equation*}5x^2+5y^2-140x-60y+900=0\end{equation}

(2)   \begin{equation*}x^2+y^2-28x-12y+180=0\end{equation*}

Two equations and two unknowns which we can solve simultaneously. Both equations are circles.

Equation (1) becomes

(3)   \begin{equation*}(x-8)^2+(y-2)^2=64\end{equation*}

and equation (2) becomes

(4)   \begin{equation*}(x-14)^2+(y-6)^2=52\end{equation*}

From equation (3)

    \begin{equation*}x=\sqrt{64-(y-2)^2}+8\end{equation}

We will worry about the negative version later.

Substitute for x into equation (4)

    \begin{equation*}(\sqrt{64-(y-2)^2}+8-14)^2+(y-6)^2=52\end{equation}

    \begin{equation*}(\sqrt{64-(y-2)^2}-6)^2+(y-6)^2=52\end{equation}

    \begin{equation*}64-(y-2)^2-12\sqrt{64-(y-2)^2}+36+y^2-12y+36=52\end{equation}

    \begin{equation*}136-y^2+4y-4+y^2-12y-12\sqrt{64-(y-2)^2}=52\end{equation}

    \begin{equation*}-12\sqrt{64-(y-2)^2}=-80+8y\end{equation}

    \begin{equation*}\sqrt{64-(y-2)^2}=\frac{20}{3}-\frac{2}{3}y\end{equation}

Square both sides of the equation

    \begin{equation*}64-(y-2)^2=\frac{400}{9}-\frac{80}{9}y+\frac{4}{9}y^2\end{equation}

    \begin{equation*}60-y^2+4y=\frac{400}{9}-\frac{80}{9}y+\frac{4}{9}y^2\end{equation}

    \begin{equation*}0=\frac{13}{9}y^2-\frac{116}{9}y-\frac{140}{9}\end{equation}

    \begin{equation*}0=13y^2-116y-140\end{equation}

    \begin{equation*}0=13y^2-130y+14y-140\end{equation}

    \begin{equation*}0=13y(y-10)+14(y-10)\end{equation}

    \begin{equation*}0=(y-10)(13y+14)\end{equation}

(5)   \begin{equation*}\therefore y=10, y=-\frac{14}{13}\end{equation*}

Substitute y=10 into x=\sqrt{64-(y-2)^2}+8

    \begin{equation*}x=\sqrt{64-(y-2)^2}+8\end{equation}

(6)   \begin{equation*}x=8\end{equation*}

Substitute y=-\frac{14}{13} into x

    \begin{equation*}x=\sqrt{64-(-\frac{14}{13}-2)^2}+8\end{equation}

(7)   \begin{equation*}x=\frac{200}{13}\end{equation*}

Now we need to consider the negative version of x. If you work through (like I did above) you end with the same equation for y.

Hence our two values for v are v=(8,10) or v=(\frac{200}{13},-\frac{14}{13}).

Would someone be expected to do this in an exam? I hope not, but I think its worth doing.

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Filed under Closest Approach, Vectors, Year 11 Specialist Mathematics