Honors Analysis
Section 6.2: Special Right Triangles
Q Problems
Q1.) The probability Mandy will correctly guess a problem on a quiz is 1/5. What is the probability she will guess 5 in a
row correctly?
Q2.) A multiple choice math contest has questions with 4 multiple choice options. A correct answer is worth a point. A
half point is subtracted for an incorrect answer. A guess is worth zero points. Zed wants to decide if it is to his
advantage to guess randomly on a question. What is the expected value of a question if Zed makes a random guess?
Q3.) Calculate the sum of the interior angles of a decagon.
Q4.) A regular polygon has interior angles measuring 20º each. How many sides does it have? (Hint: Use exterior angle!)
Q5.) Give the prime factorization of 1764 and 450. (Note: All prime factors will be single-digit values)
Q6.) Show how you can use the prime factorizations to find the Greatest Common Factor of 1764 and 45. What is it?
Q7.) Explain how to use the prime factorizations of 1764 and 450 to find the Least Common Multiple. What is the LCM?
Q8.) Write in vertex form by completing the square: 𝑦 = 3𝑥 2 − 12𝑥 + 5. Then give the vertex point.
Q9.) A price is increased by 20%. It is then increased an additional 10%. What is the percent increase? (It isn’t 30%!)
Q10.) Simplify: √36𝑥 4 𝑦 2 . (In other words, what value will multiply by itself to equal the expression under the radical?
Write each expression in simplified radical form. Rationalize all denominators.
1.) What is the conjugate of 4 − √3?
2. Write each expression in simplified radical form. Rationalize all denominators. Do NOT give answers as decimals!
a) 2√3 ∙ 4√6
b)(4√5)2
g) √32 + 42
h)
6
√3
c) (2 + √3)(6 − 2√5)
i)
3
2+√5
16
d)√25
j) √72 + √75 − √48
e)
√20
√10
k)
8+4√6
2
f) √18 + √50
3.) Complete the Pythagorean Triples. (Note that each base triple is made up of numbers that are relatively prime to
each other – that is, the only factor all three numbers have in common is 1, unlike triples such as 6, 8, 10)
3, 4, __
5, ___, 13
7, 24, ___
8, ___, 17
9, 40, ___
11, 60, ___
4.) Identify the Pythagorean Triple for each triangle. Then find the missing side.
5. Use the converse of the Pythagorean Theorem to decide if the triangle with given side lengths is acute, obtuse, or
right.
A) 11, 12, 16
B) 6, 8, 12
6.) A rhombus has diagonals of length 16 cm and 30 cm. What is the perimeter?
7.) Calculate the perimeter of an isosceles triangle with a base of 16 m and a height of 15 m.
8.) What is the diagonal of a rectangle with sides of 20 and 48 inches?
9.) Find the length of the upper base of
the isosceles trapezoid:
10.) Find QD:
11.)
12.) Calculate the length of the diagonal of a
3 x 4 x 12 rectangular prism.
13.) Write a coordinate proof to show that the diagonals of all rectangles are congruent. (Remember: you must use
variable values for the vertices!)
14.) A submarine travels an evasive course, trying to outrun a destroyer. It travels 1 km north, then 1 km west, then 1
km north, then 1 km west, and so forth, until it has traveled a total of 41 km. How many kilometers is the sub from the
point at which it started?
15.) The Rule of Pythagoras is a method that generates sets of whole numbers representing sides of right triangles.
Using the diagram below, prove algebraically that the rule does indeed generate sides of a right triangle:
ACT Practice
16.)
20.)
17.)
18.)
21.)
Honors Analysis
Enrichment Topic #1A:
Introduction to Imaginary Numbers
If you tried to solve the equation 𝑥 2 + 4 = 0, you would notice that if attempted to isolate the x variable, you would get the
equation 𝑥 2 = −4. Normally you would take the square root of both sides, but in this case, you can’t take the square root of a
negative… or CAN you??
Imaginary numbers involve the definition of a new number i, which is defined as √−1. Most importantly, if the number 𝑖 = √−1,
then the SQUARE of i would equal -1, right?? This results in “magical” numbers that CAN be squared to equal negatives. For
example, 5i x 5i = 25i2, but since i2 = -1, this means that 5i x 5i = -25!!! So now we have numbers that can be squared to result in
negatives! Any time you want to take the square root of a negative, you can represent it using imaginary numbers.
Here’s how imaginary numbers work: Let’s say you want to write √−16 using imaginary numbers. The key in all cases is to break up
the number as a product of √−1 and a second radical. In this case, √−16 = √16 ∙ √−1. Both of these can be simplified, since
√16 = 4 and √−1 = 𝑖, resulting in the value 4i. If you wanted to square 4i, you would get 16i2, which is the same thing as -16.
This also works with negative values that don’t include perfect squares. For example, √−18 breaks down into √18 ∙ √−1, which
breaks down into √18 𝑖 (often written as 𝑖√18 so the i does not appear to be under the radical). Then √18 can be simplified as
3√2, so √18 𝑖 simplifies as 3√2 𝑖, often written as 3𝑖√2.
Simplify the following values using the imaginary number i:
E1.) √−81
E2.) √−20
E3.) 2𝑖 ∙ 3𝑖
E4.) √−4 ∙ √−100
E5.) √−17
E6.)
4+√−12
2
Complex numbers (often abbreviated with an italic C) are numbers with both a real and an imaginary part. An example is 4 + 5i.
Complex numbers can be added or subtracted just by combining like terms, resulting in new complex numbers: (3 + 5i) + (7 – 2i) = 10
+ 3i. The operations of addition and subtraction are said to be CLOSED under the set of complex numbers because the sum or
difference will always result in another complex number (assuming that 4 + 0i, for example, is still considered complex). Complex
numbers can be multiplied using FOIL, but some simplification is required:
(2 + 3𝑖)(4 − 5𝑖) = 8 − 10𝑖 + 12𝑖 − 15𝑖 2 , which simplifies to 8 + 2𝑖 − 15𝑖 2. Since i2 = -1, this can further simplify down to
8 + 2i + 15, which simplifies to 23 + 2i, another complex number! So multiplication is also closed under the set of complex numbers.
Simplify:
E7.) (4 + 3i) + (-2 +10i)
E8.) (7 – 3i) – (5 – 4i)
E9.) (2 + 3i)(4 + 6i)
E10.) (4 + 3i)2
E11.) Solve for both values of x using imaginary numbers: 2x2 – 4 = -40
E12.) Use the quadratic formula to find the
complex solution to the quadratic
𝑥 2 − 10𝑥 + 34 = 0
E13.) The Fundamental Theorem of Algebra suggests than an n-degree
polynomial will have as many as n real or imaginary roots (solutions).
Factor and then use imaginary numbers to find all four solutions:
𝑥 4 − 81 = 0