Algebra/Chapter 14/Exercises

A set of exercises related to concepts from Chapter 14.

This set contains 14 exercises (0 Conceptual Questions + 14 Exercises + 0 Projects)

Exercises

Section 14.1

(★) 14.1 (Pythagorean Triples) Determine if the following sets of numbers are Pythagorean Triples.

1. (5, 12, 13)
2. (9, 60, 61)

(★) 14.2 (Using the Pythagorean Theorem) Find the missing lengths of the following right triangles.

(★) 14.3 (Diagonal of a Square) What is the length of the diagonal of a square whose side length is:

1. 2
2. 3
3. 4
4. 5
5. r

(★) 14.4 (Diagonal of a Rectangle) What is the length of the diagonal of a rectangle whose side lengths are:

1. 1 and 2
2. 3 and 5
3. 4 and 7
4. r and 2r
5. 3r and 5r
6. 4r and 7r

(★★) 14.5 (Diagonal of a Cube) What is the length of the diagonal of a cube whose side length is:

1. 2
2. 3
3. 4
4. 5
5. r

(★★) 14.6 (Diagonal of a Rectangular Prism) What is the length of the diagonal of a rectangular prism whose side lengths are:

(★★) 14.7 (Diagonal of a Rectangular Prism II) What is the length of the diagonal of a rectangular prism whose side lengths are a, b, and c? What if the side lengths were ra, rb, and rc instead?

(★★) 14.8 (Using the Pythagorean Theorem) A right triangle has the side lengths $x+1$, $x-1$, and ${\frac {x}{2}}$. Find $x$.

(★★) 14.9 (Pencil Box) A pencil box, in the shape of a rectangular prism, measures 15 cm by 12 cm by 9 cm. Find the length of the longest pencil that can fit inside.

(★★) 14.10 (Equilateral Triangle) Find the area of an equilateral triangle with 2 cm sides.

(★★) 14.11 (Regular Hexagon) Find the area of a regular hexagon with 4 cm sides.

(★★) *14.12 (Euclid's Formula) We can construct a set of Pythagorean Triples through a relation called Euclid's Formula.

When we define m and n as positive integers, and m > n:

• $a=m^{2}-n^{2}$
• $b=2mn$
• $c=m^{2}+n^{2}$

Use the Pythagorean Theorem to prove Euclid's Formula.

Section 14.2

(★) 14.13 (Right Triangle Trigonometry) Find the exact values of the six trigonometric functions of $\theta$.

Section 14.5

(★★) *14.14 (Sector Area) If ${\frac {s_{1}}{r_{1}}}={\frac {s_{2}}{r_{2}}}$, prove that $\theta _{1}=\theta _{2}$.