Honors Analysis
Section 6.3: Special Right Triangles
Q Problems
Q1.) Simplify: 4−2
Q3.) Simplify: √25𝑎4 𝑏 6
1
3
Q2.) Simplify: 125
Q5.) Simplify: √𝑥 2 + 10𝑥 + 25. If you are stuck,
3
Q4.) √27𝑎6 𝑏 3
Q6.) Rationalize the denominator:
5
4−√6
think about what a square root means, and then try
factoring the expression.
Q7.) Determine the volume
of a cylinder with a height of
Q8.) Calculate the lateral surface area
of the cylinder in Q7.
8 cm and a 5 cm radius.
Q9.) Determine the altitude of a
square pyramid with a 10 x 10 base
and a slant height of 13 units.
Q10.) What is the volume of the pyramid in Q9?
1.) Divide a square with dimensions x by x into
halves along the diagonal to form 45-45-90
triangles. Then use the Pythagorean Theorem
to derive the relationships between the sides.
2.) Draw an equilateral triangle with side lengths
of 2x. Split the triangle into two 30-60-90 triangles
using an altitude. Then use the Pythagorean Theorem
to derive the relationships between the sides.
3.) Determine the lengths of the missing sides of the 45-45-90 triangles shown:
4.) Determine the lengths of the missing sides of the 30-60-90 triangles shown:
5.) Determine each missing side length. Write all values in simplified radical form. Rationalize all denominators.
6.) Find the area of an equilateral triangle with
4 cm sides.
7.) What is the area of an isosceles triangle with a base
length of 18 ft and a vertex angle measuring 120º?
8.) An equilateral triangle with a 24 cm altitude has
what area?
9.) One of the angles of a rhombus measures 120º. If
the perimeter is 24 in, find the length of each diagonal.
10.) What is the slope of a line passing through the origin
that forms a 30º angle with the x-axis?
11.) What is the slope of a line passing through the
origin that forms a 60º angle with the x-axis?
12.) Calculate the area of the isosceles trapezoid:
13.) Determine, to the nearest tenth, the perimeter of
the trapezoid.
14.) The “span” of a hexagon is the distance
between opposite sides. Determine the span
of the regular hexagon shown:
15.) Determine the perimeter and the span of the
regular octagon shown. (Hint: Calculate angle
measures within the octagon.)
16.)
17.)
18.)
19.)
20.) Write an algebraic proof demonstrating that an
equilateral triangle with sides of length s has an
21. A) What is the length of the diagonal of a
5 x 5 x 5 cube?
area of
𝑠2
√3
4
B) What is the length of the diagonal of an
x by x by x cube?
Honors Analysis
Enrichment Topic #1B: Imaginary Numbers
Powers of i:
You already know that the imaginary number i is defined as √−1, and that as a result, i2 = -1. Other powers of i can also
be determined by writing the numbers using values that are already known. For example, i3 can be written as 𝑖 ∙ 𝑖 2 .
Since i2 = -1, 𝑖 ∙ 𝑖 2 simplifies down to 𝑖 ∙ −1, or - i. Likewise, i4 can be written as the product 𝑖 2 ∙ 𝑖 2 , which is the same as
−1 ∙ −1, which means that i4 = 1. The fact that i4 = 1 is very useful, because all additional powers of i can be written as
the product of i4 and a second value, which makes for easy simplifications.
For example, 𝑖 5 = 𝑖 4 ∙ 𝑖. Since i4 = 1, the expression simplifies down to i. 𝑖 6 = 𝑖 4 ∙ 𝑖 2 = 1 ∙ −1 = −1. Look at the
following list of powers of i, and look for a pattern:
𝑖1 = 𝒊
𝑖 2 = −𝟏
𝑖 3 = −𝒊
𝑖4 = 𝟏
𝑖5 = 𝑖4 ∙ 𝑖 = 𝒊
𝑖 6 = 𝑖 4 ∙ 𝑖 2 = −𝟏
𝑖 7 = 𝑖 4 ∙ 𝑖 3 = −𝒊
𝑖8 = 𝑖4 ∙ 𝑖4 = 𝟏
𝑖 9 = 𝑖 4 ∙ 𝑖 4 ∙ 𝑖1 = 𝒊
𝑖 10 = 𝑖 4 ∙ 𝑖 4 ∙ 𝑖 2 = −𝟏
As you might notice, the powers of i cycle from i to -1 to –I to 1. So to calculate any power of i, the key is to cancel all
powers of i4. So for example, 𝑖 22 = 𝑖 4 ∙ 𝑖 4 ∙ 𝑖 4 ∙ 𝑖 4 ∙ 𝑖 4 ∙ 𝑖 2 = −1.
Use this pattern to calculate the following powers of i:
E1.) 𝑖 11
E2.) 𝑖 17
E3.) 𝑖 31
E4.) 𝑖 101
E5.) 𝑖 120
E6.) (2𝑖)6
Division of Complex Numbers
Similar to rules for radicals, it is considered bad form to have an imaginary number in the denominator of a fraction.
Similar to radicals, an i in the denominator can be squared to result in a real number value, but the numerator must also
4
be multiplied by the same value. So 𝑖 can be written in a more standard form by multiplying the numerator and
4 𝑖
4𝑖
denominator by i: ∙ =
= −4𝑖. If the denominator involves addition or subtraction, the numerator AND
𝑖 𝑖
−1
denominator can be multiplied by the conjugate of the complex number – that is, the complex number with the sign of
the imaginary part switched.
Example:
5
2+𝑖
→
5
2−𝑖
∙
2+𝑖 2−𝑖
5(2−𝑖)
= (2+𝑖)(2−𝑖) =
10−5𝑖
4−2𝑖+2𝑖−𝑖 2
E7.) What is the conjugate of 3 + 2i?
E9.) Divide:
6
3−𝑖
E10.) Divide:
=
10−5𝑖
4−𝑖 2
=
10−5𝑖
4−−1
=
10−5𝑖
5
=
2−𝑖
1
=2−𝑖
E8.) What is the product of 4 – 5i and its conjugate?
4
5+2𝑖
E11.) Divide:
3+4𝑖
1−2𝑖