The polygon below was once a rectangle with sides 10 and 14 before a triangle was cut off. What is the perimeter, in inches, of this polygon?

Solution
To solve this problem, first determine the length of the unknown side of the polygon. To do so, use the fact that the piece cut off was a right triangle. The unknown side is the hypotenuse of the triangle with legs 14−10 = 4 and 10 − 7 = 3. Use the Pythagorean Theorem to calculate the length of the hypotenuse (the unknown side of the polygon):
4^{2} + 3^{2} = c^{2}
16 + 9 = c^{2}
25 = c^{2}
c = 5The length of the unknown side is 5. Therefore the perimeter of the entire polygon is 14 + 10 + 10 + 7 + 5 = 46.
For a ≠ 0, \(\frac{a^{9}}{a^{3}}\) is equivalent to:

Solution
The expression \(\frac{a^{9}}{a^{3}}\) is simplified by subtracting the exponents to get \(\frac{a^{9}}{a^{3}}\) = a^{9−3} = a^{6}.
If x^{2} − y^{2} = 49 and x − y = 7, then x =?

Solution
To solve for x, substitute y = x − 7 into the equation x^{2} − y^{2} = 49 (Since x − y = 7, y = x − 7):
x^{2} − (x − 7)^{2} = 49
x^{2} − (x^{2} − 14x + 49) = 49
x^{2} − x^{2} + 14x − 49 = 49
14x − 49 = 49
14x = 98
x = 7
Each of the 3 lines crosses the other 2 lines as shown below. Which of the following relationships, involving angle measures (in degrees) must be true?
I. m∠2 + m∠7 + m∠12 = 180°
II. m∠4 + m∠5 + m∠10 = 180°
III. m∠3 + m∠8 + m∠11 = 180°

Solution
To solve this problem, recognize that any three angles that are part of a triangle add to 180°.
Relationship I: ∠2, ∠7, and ∠12 make up the three angles within the triangle pictured, therefore m∠2 + m∠7 + m∠12 = 180°.
Relationship II: using the property that vertical angles are congruent ∠4 ≅ ∠2, ∠5 ≅ ∠7, and ∠10 ≅ ∠12, therefore m∠4 + m∠5 + m∠10 = 180°.
Relationship III: two of these angles in this relationship are clearly greater than 90° each, thus m∠3 + m∠8 + m∠11 > 180°.
Therefore the only relationships that must be true are I and II.
How many different positive 3digit integers can be formed if the three digits 3, 4, and 5 must be used in each of the integers?

Solution
To easily solve this problem, write out every possible threedigit integer: 345, 354, 435, 453, 534, and 543; six different positive threedigit numbers can be formed.
How many ordered pairs (x, y) of real numbers will satisfy the equation 5x − 7y = 13?

Solution
The equation 5x − 7y = 13 defines a line. Since there are an infinite number of points in a line, there are an infinite number of ordered pairs (x, y) of real numbers that satisfy the equation 5x − 7y = 13.
In the figure below, what is the sum of p and q?

Solution
To solve this problem, set up three other variables, x, y, and z. Let x be the supplement of angle p.
Let y be the supplement of angle q.
Let z be the angle opposite of the 105° angle, so z = 105°. With these new variables, you know that p = 180 − x and q = 180 − y.
This means that p + q = (180 − x) + (180 − y), which simplifies to 360−(x+y). You can see that the three lines form a triangle and that one of the angles is 105°.
That means that the other two angles, x and y must have a sum of 180 − 105, or 75°.
Thus, if x+y = 75°, then p+q = 360−(x+y), or 360−75,which is 285°.
If \(\sqrt{2x}\) + 5 = 9, then x = ?

Solution
To solve this problem,first subtract 5 from both sides of \(\sqrt{2x}\) + 5 = 9 to get \(\sqrt{2x}\) = 4. Squaring both sides of \(\sqrt{2x}\) = 4 yields 2x = 16, or x = 8.
One traffic light flashes every 6 seconds. Another traffic light flashes every 9 seconds. If they flash together and you begin counting seconds, how many seconds after they flash together will they next flash together?

Solution
To solve this problem, find the least common multiple of 6 and 9. The correct answer is 18 because 6 × 3 = 18 and 9 × 2 = 18.
Which of the following is equal to \(\frac{\left ( \frac{1}{3}\frac{1}{4} \right )}{\left (\frac{1}{3}+\frac{1}{4} \right )}\)?

Solution