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Sunday Afternoon Maths LXVICryptic crossnumber #2
Sunday Afternoon Maths LXVCryptic crossnumber #1
Square and cube endings
Sunday Afternoon Maths LXIVEqual lengths
Sunday Afternoon Maths LXIIIIs it equilateral?
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Is it equilateral?
Source: Chalkdust issue 07
In the diagram below, \(ABDC\) is a square. Angles \(ACE\) and \(BDE\) are both 75°.
Is triangle \(ABE\) equilateral? Why/why not?
Earlier this year, I wrote a blog post about different ways to prove Pythagoras' theorem. Today's puzzle uses Pythagoras' theorem.
Start with a line of length 2. Draw a line of length 17 perpendicular to it. Connect the ends to make a right-angled triangle. The length of the hypotenuse of this triangle will be a non-integer.
Draw a line of length 17 perpendicular to the hypotenuse and make another right-angled triangle. Again the new hypotenuse will have a non-integer length. Repeat this until you get a hypotenuse of integer length. What is the length of this hypotenuse?
Source: New Scientist Enigma 1773
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?
Source: Maths Jam
The three sides of this triangle have been split into three equal parts and three lines have been added.
What is the area of the smaller blue triangle as a fraction of the area of the original large triangle?
Equal side and angle
Source: Jim Noble on Twitter
In the diagram shown, the lengths \(AD = CD\) and the angles \(ABD=CBD\).
Prove that the lengths \(AB=BC\).
Source: Futility Closet
Prove that \(\arctan(1)+\arctan(2)+\arctan(3)=\pi\).
Source: Futility Closet
This unit square is divided into four regions by a diagonal and a line that connects a vertex to the midpoint of an opposite side. What are the areas of the four regions?