Geometry Problem

Test Preparation SAT Preparation
1

<p>Came across a question in CB's 2007-2008 practice PSAT, and it was the only math question that I couldn't struggle myself through. I did quick sketch of the picture included with the problem, you can see it here:</p>

<p>ImageShack</a> - Hosting :: geometryproblemuf3.jpg</p>

<p>Yes a realize it's bad :P.</p>

<p>Well the question says that P is the center with arc length 6, r's are the radii, and the two radii are extended by 3. Find X.</p>

<p>I know the answer is (6r+18)/r, but I have no idea how to get there. A formula/explanation would help a lot!</p>

2

<p>okay to add one additional variable I'm going to make:
angle measure = A</p>

<p>okay first for the smaller arc you create an expression: (2<em>pi</em>r)/a=6
solving for that gives you (pi*r)/3=a</p>

<p>Then you can create an equation using the larger arc : (2<em>pi</em>(r+3))/a=x
Substitute a with the previous equation: (2<em>pi</em>r)/((pi*r)/3)=x
simplify to get:6(r+3)/r</p>

<p>which when expanded gives you 6r+18/r. Hope that helped.</p>

3

<p>What I did was find what fraction 6 was of the inner circle, which was
6/2(pi)r
I then found the fraction x was of the big circle, which was
x/(2(pi)r + 6pi) </p>

<p>Set the two equations equal together and solve for x, because the fractions should be equal to each other...</p>

<p>x/(2(pi)r + 6pi) = 6/ 2(pi)r</p>

<p>Solve for x</p>

4

<p>you could solve the whole thing in radian which would make it a bit more simpler.
First
the angle measure=x(in radian)
2pi angle=2pir
x angle=xr
xr=6
x=6/r
Similarly
larger arc=x(r+3)
Substituting the value of x:
x(r+3)=6(r+3)/r
=(6r+18)/r</p>

5

<p>Since circumference is directly proportional to radius, the ratio of two circles' circumferences is equal to the ratio of their radii (and diameters as well).</p>

<p>Arcs subtending congruent central angles in two circles constitute the same fraction of each circle (if the central angle in your diagram was 90 deg, each arc would be 1/4 of the corresponding circle).</p>

<p>Thus the ratio of such arcs is also equal to the ratio of the radii:
x/6 = (r+3) / r</p>

<h1>x = (6r+18)/r</h1>

<p>Now, what could really help us in this question is the fact that the measure of the central angle is not given. You can arbitrarily assume any value: 30 deg, or 90 deg, or (why not?) - 360 deg.
In the latter case each arc becomes a full circle, and the ratio of their lengths
x/6 = (2 pi (r+3)) / (2 pi r).</p>

<h1>Same difference. :)</h1>

<p>Arbitrarily assuming values is a very powerful technique which works wonders in many SAT math questions.</p>

6

<p>Thanks gcf, your explanation seemed like it provided the fastest way to answer the problem.</p>

<p>I guess not having the angle measurement through me off, at least I can solve it now.</p>

<p>Thanks a lot for every one's input!</p>