Equal Lengths
Weekly Problem 29 - 2013
An equilateral triangle is drawn inside a rhombus, both with equal side lengths. What is one of the angles of the rhombus?
An equilateral triangle is drawn inside a rhombus, both with equal side lengths. What is one of the angles of the rhombus?
Image
![Equal Lengths Equal Lengths](/sites/default/files/styles/large/public/thumbnails/content-id-2846-Equal%252520lengths%252520Stefania.png?itok=YmnmkPIN)
The diagram shows an equilateral triangle inside a rhombus. The sides of the rhombus are equal in length to the sides of the triangle.
What is the value of $x$?
Image
![Equal Lengths Equal Lengths](/sites/default/files/styles/large/public/thumbnails/content-id-2846-Equal%252520lengths%252520sol%252520Stefania.png?itok=98ozXQTb)
Triangle $DFC$ is isosceles $(CF=CD)$.
Hence $\angle DFC = \angle FDC = x^{\circ}$.
Hence $\angle FCD = (180-2x)^{\circ}$ (angle sum of triangle).
Now $\angle EBC = \angle FDC = x^{\circ}$ (opposite angles of a parallelogram) and triangle EBC is isosceles (CE = CB). Hence $\angle BEC = x^{\circ}$ and $\angle ECB = (180-2x)^{\circ}$
Lines $AD$ and $BC$ are parallel and hence $$\angle ADC + \angle BCD = 180^{\circ}$$ Therefore: $$x+2(180-2x) + 60= 180$$i.e.$420-3x=180$ i.e. $3x=240$ i.e. $x=80$