Triangle ABC is not equilateral, and angle ABC = 60 degrees

Quantity A	Quantity B
The angle opposite the shortest side of the triangle	60 degrees

Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.

Solution

The smallest angle in a triangle is always opposite the shortest side. If angle ABC is 60 degrees, the other two angles total 180° – 60° = 120°.The triangle isn’t equilateral, the remaining two angles cannot both be 60°.Therefore, the smaller angle must be less than 60°, and Quantity B is greater.

Solution

C, D,E, and F

The Third Side Rule states that the third side of any triangle must be greater than the difference between the other two sides and less than the sum of the other two sides.Therefore, the third side of the triangle in the question must be between 4 and 12, and you can eliminate any choices outside this range.The only choices in this range are 5, 6, 7, and 8, the correct answers.

A hiker left her tent and traveled due east for 5 miles, then traveled due south for 24 miles, then due east for 5 miles, arriving at a hut. What is the straight-line distance from her tent to the hut?

Solution

First, draw the picture (see below). Notice that this makes two right triangles, each with legs of 5 and 12.Either recognize the 5-12-13 triple or use the Pythagorean theorem to see that the distance is 13 + 13 = 26.

In triangle ABC, side AB has a length of 12, and side BC has a length of 5.

Quantity A	Quantity B
The length of side AC	7

Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given

Solution

The Third Side Rule states that the third side in any triangle must be shorter than the sum of, and longer than the difference between, the other two sides. Hence, the third side of this triangle must be greater than 7, and less than 17. Quantity A is greater.

What is the area of the rectangle shown above?

Solution

Recognize the 3-4-5 triple or use the Pythagorean theorem to find that the missing side length of the rectangle is 4.The area of the rectangle is bh = 3 × 4 = 12, so the answer is choice (E).

A ship captain sails 500 miles due south and then 1,200 miles due east.

Quantity A	Quantity B
1,350 miles	The minimum number of miles the captain must sail to return to his original position

Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.

Solution

Draw a right triangle representing the captain’s route so far and the path back to his starting point:

A right triangle with legs of 500 and 1,200 is a multiple of the familiar 5-12-13 triangle, so the hypotenuse—and the number of miles the captain must sail to return to his original position—is 1,300.The answer is choice (A).

The length of line segment AC is ³⁄₄ the length of line segment AB.

Quantity A	Quantity B
BC	6

Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.

Solution

AC has a length of 3, so you can use Pythagorean theorem, or recognize the Pythagorean triple, to find that BC has a length of 5.The answer is choice (B).

In the figure above, what is the value of \(\frac{a+b+c}{30}\)?

Solution

All three angles of the triangle add up to 180°. 30 goes into 180 six times.The answer is choice (B).

In square ABCE, AB = 4.

Quantity A	Quantity B
24	The perimeter of polygon ABCDE

Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.

Solution

∆CDE has equal angles, so it is equilateral. ABCE is also equilateral, as are all squares.To find the perimeter of any figure, add up all of the side lengths on the outside of the figure. In this case, 5 equal segments of length 4 result in a perimeter of 20, so Quantity A is greater.

In the figure above, WX = XY and points W,Y, and Z lie on the same line. What is the value of q?

Solution

105

There are 180 degrees in both a straight line and a triangle. In the figure, ∠XWY and ∠XYW are congruent and their measures add up to 180° − 30° = 150°, so each angle measures 75°. A straight line measures 180°, so q = 180 − 75 = 105.

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