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'Can You Explain Why?' printed from http://nrich.maths.org/
Kkytha (Wimbledon High School) provided us
with a nice explanation:
"Diagram 2 (D.2) shows the shape reflected across its horizontal
Diagram 3 (D.3) is Diagram 2 rotated 90° clockwise. You
can see that the angle of the tip is 30°+30°,
adding up to 60°. If a line is drawn between the bottom
two vertices as shown in blue, that line should be 12cm because the
other two sides are 12cm and the angle between them is
60°(making the triangle an equilateral triangle). The
blue line drawn should form an isosceles triangle with the two 5cm
lines. In an isosceles triangle, the length of the two identical
sides must add up to greater than the length of the base. In this
particular isosceles triangle, the identical sides add up to 10cm
which is shorter than the base of 12cm. Thus, the triangle in the
problem is impossible to construct!"
Giancarlo used trigonometry to solve the
problem. By extending the original triangle to make a right-angled
triangle, he found that the length of the new, shorter side was
12sin30 cm, which is 6cm, and so there is no solution for the same
reason as above.