Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

enter image description here

I thought the answer would be square root of 3. It would seem that the x co-ordinate of Q would just be the opposite of the x co-ordinate of P.

I'm not sure if the picture is just being deceptive, or if I just don't remember my math from high school very well...

However... I'm told the correct answer is 1.

I don't understand why... Could someone explain that please?

Thanks so much!

share|cite|improve this question
Someone is sure to leave a great answer shortly, so I'm just gonna drop this comment and say that when you rotate by 90 degrees, you swap the x and y coordinates and then negate one of them, depending on which direction you rotated. – El'endia Starman Aug 16 '11 at 1:26
@El'endia Starman: Seems like a good clear answer to me, with the addition of a little detail. – André Nicolas Aug 16 '11 at 1:29
it just seems to me that along the x axis, the point P is as far from the origin as the point Q... But maybe just because the picture is not drawn to scale? – kralco626 Aug 16 '11 at 1:29
That is indeed the case; the picture is not quite correct. That tends to happen with most images on tests/homework, actually... – El'endia Starman Aug 16 '11 at 1:31
it's actually a question on an "official" GMAT preparation test, for which the directions say "All figures are drawn as accurately as possible. Exceptions will be clearly noted"... but anyways I now understand and will hopefully get it right on the actual test! – kralco626 Aug 16 '11 at 1:35
up vote 5 down vote accepted

A simple answer to your question is that when you rotate by $90$ degrees (as indicated by the right angle symbol), you swap the $x$ and $y$ coordinates and then negate one or the other, depending on which direction you rotated it. In your case, you had $(-\sqrt{3}, 1)$, which became $(1, -\sqrt{3})$ and then because you rotated into the first quadrant, the final point is $(1, \sqrt{3})$.

share|cite|improve this answer
Thanks for the quick reply! – kralco626 Aug 16 '11 at 1:39
@kralco626: You're welcome. :) – El'endia Starman Aug 16 '11 at 1:41

There are various "high school" approaches to the answer. You will have to help me by drawing a picture.

Drop a perpendicular from the point $P$ to a point $M$ on the negative $x$-axis. Look at the angle $MOP$, and call it $\theta$. In $\triangle OPM$, the hypotenuse $OP$ has length $\sqrt{(\sqrt{3})^2+1}$, which is $2$. Thus $\sin\theta=1/2$.

You may recall "special angles." The angle $\theta$ between $0^\circ$ and $90^\circ$ such that $\sin\theta=1/2$ is the $30^\circ$ angle.

Now drop a perpendicular from $Q$ to the point $N$ on the positive $x$-axis. Let $\phi$ be the angle $QON$. What is the size of $\phi$? It is $180^\circ-(90^\circ+30^\circ)$, which is $60^\circ$. Thus $\phi$ is a lot bigger than the $30^\circ$ angle $\theta$, so the picture should not be at all symmetrical bout the $y$-axis!

Note that the cosine of the angle $\phi$ is $s/2$. But the cosine of the $60^\circ$ angle is $1/2$. It follows that $s=1$.

Without special angles: Do the constructions of $M$ and $N$ exactly as in the first solution, and let $\theta$, $\phi$ be as described there.

Note that $\theta+\phi=90^\circ$, so $\theta$ and $\phi$ are complementary angles.

Now compare $\triangle OPM$ and $\triangle QON$. We have $OP=QO$, and $\angle OPM=\angle QON$. So the triangles are congruent. Note that $PM=ON$. But $PM=1$, and therefore $s=ON=1$.

Using some analytic geometry: The slope of the line $OP$ is $-1/\sqrt{3}$. But $OQ$ is perpendicular to $OP$, so its slope is the negative reciprocal of $-1/\sqrt{3}$. Thus the slope of $OQ$ is $\sqrt{3}/1$.

But the slope of $OQ$ is $t/s$. It follows that $t=s\sqrt{3}$. By the Pythagorean Theorem, $s^2+t^2=4$. So $s^2+3s^2=4$. Since $s$ is positive, we conclude that $s=1$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.