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)?
Area of △RAC = ½ah
By dropping a perpendicular (h), from C to RA, we create two right triangles: △RDC and △DAC.
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.
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 = h√3.
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 = h√3.
Putting together this knowledge, we can say that:
a = RA = RD + DA = h√3 + h = h(√3 + 1)
So, 
By substituting this back into Area of △RAC = ½ah, we get:
How about the area of △GCM?