Sunday Afternoon Maths VLIII
Cutting corners
The diagram below shows a triangle \(ABC\). The line \(CE\) is perpendicular to \(AB\) and the line \(AD\) is perpedicular to \(BC\).
The side \(AC\) is 6.5cm long and the lines \(CE\) and \(AD\) are 5.6cm and 6.0cm respectively.
How long are the other two sides of the triangle?
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Let \(A\), \(B\) and \(C\) represent the angles at points \(A\), \(B\) and \(C\). Looking at triangle \(ACE\) gives:
$$\sin A=\frac{5.6}{6.5}$$
And triangle \(ACD\) gives:
$$\sin C=\frac{6.0}{6.5}$$
Using Pythagoras' Theorem to find the missing sides in these triangles also gives:
$$\cos A=\frac{3.3}{6.5}$$
$$\cos C=\frac{2.5}{6.5}$$
We can use these to find \(\sin B\):
$$\begin{array}{rl}
\sin B&=\sin(\pi-A-C)\\
&=\sin(A+B)\\
&=\sin A\cos C + \cos A\sin C\\
&=\frac{5.6}{6.5}\frac{2.5}{6.5}+\frac{6.0}{6.5}\frac{3.3}{6.5}\\
&=\frac{33.8}{6.5^2}
\end{array}$$
Now the missing sides can be found using the sine rule:
$$\begin{array}{rl}
CB&=\frac{CA\sin A}{\sin B}\\
&=\frac{6.5\times\frac{5.6}{6.5}}{\frac{33.8}{6.5^2}}\\
&=\frac{6.5^2\times5.6}{33.8}\\
&=7\text{cm}
\end{array}$$
$$\begin{array}{rl}
AB&=\frac{CA\sin C}{\sin B}\\
&=\frac{6.5\times\frac{6}{6.5}}{\frac{33.8}{6.5^2}}\\
&=\frac{6.5^2\times6}{33.8}\\
&=7.5\text{cm}
\end{array}$$
Rugby scores
In a rugby (union) match, 3 point are scored for a kick, 5 for a try and 7 for a converted try. This scoring system means that some total scores can be achieved in different combinations, while others can be achieved in only one way.
For example, 14 can be scored in two ways (three kicks and a try; or two converted tries), while 8 can only be achieved in one way (try and a kick).
What is the highest score which can only be made in one way?
What is the highest score which can be made in two ways?
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11 is the highest score which can only be made in one way (try and two kicks).
Each of 12, 13 and 14 can be made in two ways:
- 12 - try and converted try; or four kicks
- 13 - converted try and two kicks; or two tries and a kick
- 14 - three kicks and a try; or two converted tries
Now any score higher than 14 can be made by adding kicks to one of these three. This gives at least two ways of making every score higher than this.
By a similar argument, it can be shown that 16 is the highest score which can only be made in two ways.
For the highest score which can be made in \(n\) ways, see the OEIS sequence
A261155.
Extension
Which is the highest rugby score such that if the number of tries scored is known, you can always tell how many of these tries were converted?
Quarter circle
Source: Maths Jam
A quarter circle is drawn in a square. A rectangle is drawn in the corner of the square which touches the circle and has sides of length 8 and 1.
What is the length of a side of the square?
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Let the radius of the circle by \(r\). Draw in the following two lines.
The sides of the triangle formed are \(r\), \(r-1\) and \(r-8\). By Pythagoras' Theorem:
$$(r-1)^2+(r-8)^2=r^2$$
$$r^2-2r+1+r^2-16r+64=r^2$$
$$r^2-18r+65=0$$
$$(r-13)(r-5)=0$$
$$r=5\text{ or }13$$
\(r\) must be larger than 8, \(r=13\).
![](javascript:showlimage("cut-corn.png"))
![](javascript:showlimage("quarter-circle.jpg"))
![](javascript:showlimage("quarter-circle-ans.jpg"))