Math problem

Test Preparation SAT Preparation
1

<p>This is derived from an SAT problem, though slightly modified. Same exact concepts though and technically you need to be able to do this to do the problem.</p>

<p>This is a picture from an SAT problem and I just labeled the vertices. <a href="http://s24.postimg.org/z2x3bqzzp/Problem.png%5B/url%5D"&gt;http://s24.postimg.org/z2x3bqzzp/Problem.png&lt;/a>
Prove that AB = AC.</p>

2

<p>I don’t think AB and AC must have the same length. Suppose we fix points A, B, D. The locus of points that C could be forms a circle passing through A, D, and C such that angle ACD = x. By picking points to represent C, we see that AB and AC are not always equal. Basically, you can draw the diagram such that angles ABD, ACD are equal but AB and AC are not equal.</p>

3

<p>If you simultaneously fix A, D, and B, how could the circle representing the locus of points possible go through any of the points A, D, B? We clearly see that AC, DC, and BC are all not equal to 0 (surely this much we can assume).</p>

<p>I may be wrong but If you fix points A, B, and D, then C also becomes fixed because shifting C would change the angle y, would it not?</p>

4

<p>BTW, do you attend MIT?</p>

5

<p>To see why AB and AC are not always equal, draw the circumcircle of triangle ACD. Move point C along the major arc AD. The angle ACD is still equal to x, but AC no longer equals AB. This is most easily seen when C is far away or close to point D.</p>

<p>You can also show by law of sines that AB = AC iff angles ADC and ADB are congruent, which occurs iff AD bisects angles BAC and BDC. Can’t prove that.</p>

<p>Yes, I’m a sophomore at MIT.</p>

6

<p>I believe that they can be equal. I had done similar questions last year in school which is grade IX. But the thing is, I forgot. I think it can be done using congruency or using the conjecture “sides opposite equal angle …”.</p>

<p>Edit: How do we know that D will be a point on the circle? Only if D is a point on the circle then only we can get the arc AD.</p>

7

<p>They <em>can</em> be equal but they don’t have to be equal. Two very different statements.</p>

<p>The circumcircle of triangle ACD is, by definition, the unique circle passing through points A, C, and D.</p>

8

<p>Oh! my bad, I thought it was ABC.</p>

9

<p>No problem…do you now see the reason why AB and AC are not necessarily equal?</p>