If ghjk = 24 and ghkl = 0, which of the following must DO YOUR FIGURING HERE. be true?

Solution
Since ghjk = 24, none of these variables (g, h, j, or k) can equal 0. If one of them did equal 0, their product would also be 0. Thus, eliminate answer choices C and D. If g and h were both negative, their product would still be positive and therefore, you can eliminate answer choices A and B. Since ghkl = 0 and g, h, and k cannot equal 0, l must equal 0.
On Friday, a computer was priced at $800. On the following Wednesday, the price was reduced by 15%. On the following Friday, the price was further reduced by 20%. What percent of the original price was the final price?

Solution
If a price is reduced by 15%, then 85% of the original price is retained.Likewise, if a price is reduced by 20%, then 80% of the original price is retained. If a price is reduced by 15% and then again by 20%, the percent of the original price that remains is (85%)(80%) = (0.85)(0.80), or 0.68, which is equivalent to 68%.
In ΔXYZ below, \(\overline{XZ}\) is ^{7}⁄_{8} of h, the length of the altitude. What is the area of ΔXYZ in terms of h?

Solution
Since XZ = ^{7}⁄_{8}h and the area of a triangle is ^{1}⁄_{2}bh, then the area of this triangle is ^{1}⁄_{2}(^{7}⁄_{8}h)h, or\(\frac{7}{16}\)h^{2}, answer choice J.
In the figure below, FGHJ is a square and Q, R, S, and T are the midpoints of its sides. If GH = 10 inches, what is the area of QRST, in inches?

Solution
To find the area of QRST, first determine the length of its sides. To do so, use the fact that right triangles exist in every corner of FGHJ with legs that are equal to half the length of a side of FGHJ, and with hypotenuse equal to the length of a side of QRST. Since the length of a side of FGHJ is 10, the legs of the triangles have length 5. Using the Pythagorean Theorem, the length of the hypotenuse, h, is expressed by h^{2} = 5^{2} + 5^{2} = 25 + 25 = 50; h = √50. Since the length of the hypotenuse is equal to the length of a side of QRST, the area of QRST is (√50)^{2}, or 50.
If tan α = ^{x}⁄_{y}, x > 0, y > 0, and 0 < α < ^{π}⁄_{2} , then what is cos α?

Solution
To solve this problem, it might be helpful to draw a picture in which α is an angle in a right triangle with side opposite of length x and side adjacent of length y, as shown below:
The hypotenuse, h, of this triangle can be determined using the Pythagorean Theorem, as follows:
h^{2} = x^{2} + y^{2}
h =\(\sqrt{(x^{2}+y^{2})}\)
(x2 + y2)
Since cosine is the ratio of the side adjacent to the hypotenuse, cos α = \(\frac{y}{\sqrt{(x^{2}+y^{2})}}\).
The ratio of x to z is 3 to 5, and the ratio of y to z is 1 to 5. What is the ratio of x to y?

Solution
If the ratio of x to z is 3 to 5, and the ratio of y to z is 1 to 5, then ^{x}⁄_{z} = ^{3}⁄_{5} and ^{x}⁄_{z} = ^{1}⁄_{5}. The ratio ^{x}⁄_{y} =(^{x}⁄_{z})/(^{y}⁄_{z})=(^{3}⁄_{5})(^{1}⁄_{5})=^{3}⁄_{1}. Thus, the ratio of x to y is 3:1
What is the approximate distance between the points (4,−3) and (−6,5) in the standard (x, y) coordinate plane?

Solution
Which of the following inequalities defines the solution set for the inequality 23 − 6x ≥ 5?

Solution
To solve this problem, first subtract 23 from both sides of 23 − 6x ≥ 5 to get −6x ≥ −18. Then divide both sides of the inequality by −6 (flip the direction of the inequality when dividing by a negative number) to get x ≤ 3.

Solution
The diagonal of a rectangular garden is 15 feet, and one side is 9 feet. What is the perimeter of the garden?

Solution
To solve this problem, it might be helpful to draw a picture, as shown below:
If a rectangular garden has a side of 9 feet and a diagonal of 15 feet, it forms a right triangle with leg 9 and hypotenuse 15. According to the Pythagorean Theorem, the length of the other leg, x, can be determined using the following equation:
15^{2} = 92 + x^{2}
x^{2} = 225 − 81
x^{2} = 144
x = 12
The perimeter of the rectangle is therefore 2(9) + 2(12), or 42.