Reading the question carefully, "<span>He draws the diameter of the circle through center P using a straight edge. Next he opens his compass to a width equivalent to the radius of the circle." - that implies that Mateo is still in the process of drawing the circle by which the equilateral is inscribed to.
The next steps should be:
*He should draw the upper and lower arcs using the compass opened to a width equivalent to the radius of the sircle.
*Make a vertical diameter using centroid P as basis.
*Draw two right triangles facing the opposite directions using the top most point by which the vertical diameter of the circle touches the top most circumference of the circle.
Doing these steps, Mateo will be able to draw an equilateral triangle inscribed in a circle.
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O is the midpoint so both sides FO and OG need to equal each other.
3x + 17 = 7x - 15
Subtract 3x from both sides:
17 = 4x -15
Add 15 to both sides:
32 = 4x
Divide both sides by 4:
x = 32/4
X = 8
Now you have the value of X, solve for the lengths:
FO = 3x +17 = 3(8) +17 = 24 +17 = 41
OG = 7x -15 = 7(8) -15 = 56-15 = 41
Total length would be 41 + 41 = 82
You can double the product of 9 and 4 to get to the product of 9 and 8
Combinatorial Enumeration. That whole class was a rollercoaster ride of mind-blowing generating functions to prove crazy things. The exam had ridiculous questions like 'count the number of cactus trees with n vertices such that etc etc etc' and you'd do three pages of terrible terrible sums and algebra. Then your final answer would be something beautiful like n/2 and you'd breath a sigh of relief and thank the math gods.
It is 6 because 6x6x6=216
