Question number 25. A plane is covered by squares of two different sizes as shown here to the right. The larger square has a side length of A and the smaller square has a side length of B. The dashed lines intersect at an angle measuring 30 degrees. What is the value of the ratio A to B? So I have copied here a piece of the diagram which allows me to clearly label the side lengths. So the side of the smaller square here is B and the side of the larger square labeled over here that is A. And now what we want is the ratio A to B which looks like one over B over A. That's algebraically equivalent and the reason I'm writing that is I can look at my diagram over here. I can trace out a right triangle and if I label the angle here alpha then what I have is B over A is exactly the tangent of alpha. And if I can just find the measure of that angle I can compute its tangent and that would be the reciprocal of my answer. So let's do just that. We have some information about the angles here. We know that the lines intersect at a 30 degree angle so let's put 30 degrees over here. We know that one of the diagonals intercepts two vertices of the smaller square so it makes the angle of 45 degrees with the side I have labeled B. And then noticing we have an alternate interior angle over here with the following line if we extend that edge angle alpha also appears over here. So that's alpha and now I can compute its measure as 90 degrees minus 45 degrees minus 30 degrees and that comes out to 15 degrees. So what I need to compute is then the tangent of 15 degrees and using the half angle formula that's 30 degrees divided by 2 and so that gives us the sine of 30 degrees using the half angle formula divided by 1 plus the cosine of 30 degrees and that is 1 half sine of 30 degrees the cosine is root 3 over 2 and then we simplify so we have 1 half and with the common denominator here that's 2 plus root 3 all over 2. This compound fraction simplifies then to 1 over divided by 2 plus square root of 3 and remember that is the reciprocal of the answer so finally we have 2 plus root 3 here over 1 after taking the reciprocal so the ratio of A to B is 2 plus root 3 to 1 and that is exactly choice B here so we choose B as our answer.

ratio

squares

trigonometry

angle

calculation