Hard Triangle Area in Square

If GRAW is a square with side a, and triangle GMR is equilateral, then what is the area of triangle RAC (in terms of a)?




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Area of △RAC = ½ah
By dropping a perpendicular (h), from C to RA, we create two right triangles: △RDC and △DAC.

DAC
Since GA bisects a right angle, this makes the angle DAC 45°. From there we can see that △DAC is an isosceles triangle, and thus DA = h.

RAC
Since △GMR is equilateral, it follows that all interior angles are 60°, which forces angle CRD to be 30°. Thus, △RDC is a 30°-60°-90° triangle and we can use our knowledge of trigonometric ratios to deduce that RD = h3.

Putting together this knowledge, we can say that:
a = RA = RD + DA = h3 + h = h(√3 + 1)

So,

By substituting this back into Area of △RAC = ½ah, we get:

How about the area of △GCM?


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