Triangle ABC is inscribed in a circle, such that AC is dia

[AlainQ208]

Geometry guide, 5th Edition, Chp 6 PS, #4.

Q: Triangle ABC is inscribed in a circle, such that AC is diameter of the circle and angle BAC is 45'. If the area of triangle ABC is 72 square units, how much larger is the area of the circle than the area of the triangle ABC?

My problem is simple. I figured out that the radius of the circle is 6 root 2, but then don't know how after applying the formula for the area of the circle (pie r^2) to that. The solution says that it then becomes 72pie.

If you could tell me how to apply the formula of the area of a circle to 6 times the square root of 2, and how that becomes 72pie, I would really appreciate it.

[RahulS31]

6 * square root 2 -> Radius

Area of Circle = Pi*(r)^2
= Pi * (6 * square root 2)^2
= Pi * (6)^2 * (square root 2)^2
= Pi * 36 * 2
= Pi * 72
= 72pi

[tim]

Thanks, Rahul! Let us know if there are any further questions on this one.

[RonPurewal]

the above is a bit hard to read, so here's something easier to read.

really, the issue is how to square 6√2. (the pi is there from the start; i assume that's not a problem.)

just write out the multiplication if you're not sure!

(6√2)^2
= (6√2)(6√2)
= (6)(√2)(6)(√2)
= (6)(6)(√2)(√2) ...since you can multiply things in any order
= (36)(2)
= 72

[AlainQ208]

Thanks everyone!

It's funny, as I sat there and struggled with this one, I started working the problem from beg to end explaining each step to my wife until I got to 6 * square root of 2. And then it just hit me...I guess I needed to teach her/hear the logic audibly before I found the answer (which was found about 10 mins after I posted this...)

Thanks again!

[RonPurewal]

"working through the problem out loud" is a SUPER valuable technique.

the primary value is that, if you're explaining aloud, you can't bluff anymore.
i.e., when you're just thinking through a problem, you may tolerate vagueness in your understanding. "well, i kinda think i should do this... maybe this is a thing..."
on the other hand, if you have to actually explain WHY you're doing the steps ... then you have to understand, one hundred per cent, both (i) how the steps work and (ii) why you are actually doing them.

so, if you have any trouble at all with "i keep working when i don't have a clear goal", talk through the problems aloud.
obviously this isn't something you can do on test day, but it's a super valuable learning tool.