to make the big triangle pictured above on the left. If Area = bh, then for ∆TVW, bh = 45,
so bh = 90. Looking at ∆TZW, Area = = bh = 90. If the area of ΔTZW is 90, and the area of
ΔTVW is 45, then 90 − 45 = 45, so the area of ΔXYZ is also 45, so the answer is (J).
Alternatively, consider the two triangles side by side, as pictured above on the right. Given
that a and b are the legs of both triangles, to go from the 55° angle to the 125° angle requires
dropping the height and lengthening the base proportionally. Because the base and height are
inversely related this way, the area will stay constant even as the height and base shift.
57 . E The first decision you have to make is which one (or both) of these laws is useful to you.
You’re trying to solve for a side where you have the opposite angle, but you don’t have
angles to match up with either of the other two sides you know. The sides are not equivalent,
so you can’t assume that the angles are equivalent. Consequently, you may not have enough
information to use the law of sines. The law of cosines, on the other hand, would let you
solve for the missing side, c, knowing only the other sides and the opposite angle. Line up
each piece of the formula to find that LJ^2 = 9^2 + 6^2 − 2(9)(6) cos 50°. Before you start
calculating this value, glance at your answer choices—they aren’t asking you to solve
completely, just to match up the filled-in formula. Take the square root of both sides to find
LJ = or (E).
58 . G Break this problem up into little pieces. An arithmetic sequence has a common difference
between terms, so the same number is being added to get from one term to the next. If the 7th
term is 13.5 and the 11th term is 18.3, the 9th term must be exactly halfway between them at
15.9. Halfway between the 7th term and the 9th term is the 8th term, which would be at 14.7.
If the 7th term is 13.5 and the 8th term is 14.7, then the common difference between terms is
1.2. Working backward from the 7th term, the 6th term is 12.3, the 5th term is 11.1, the 4th
term is 9.9, the 3rd term is 8.7, the 2nd term is 7.5, and the 1st term is 6.3. Adding up the first
three terms, 6.3 + 7.5 + 8.7 = 22.5, which is (G). There is a formula for arithmetic
sequences, but when you’re dealing with relatively small numbers in relatively small
quantities, sometimes the most reliable thing to do is simply write it out.
59 . C If the only possible value for w is 8, then the quadratic in factored form is (w − 8)(w − 8) = 0.
Expanding this by FOILing gives you w^2 − 16w + 64 = 0. Line this up with the original
equation to find that p must be 16, and q must be 64. With your answer choices, make sure
you know what you’re looking for—if you picked (E), you may have solved for q instead of
p.
