Chapter 6

The converse of the Pythagorean theorem is also true. It states, "If the measures \(a, b\), and \(c\) of the sides of a triangle are such that \(a^{2}+b^{2}=c^{2}\), then the triangle is a right triangle with \(a\) and \(b\) the measures of the legs and \(c\) the measure of the hypotenuse." Use the converse of the Pythagorean theorem to determine which of the triangles with sides of the following measures are right triangles. (a) \(9,40,41\) (b) \(20,48,52\) (c) \(19,21,26\) (d) \(32,37,49\) (e) \(65,156,169\) (f) \(21,72,75\)

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{-5 i}{2-4 i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{-2+6 i}{3 i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{-4-7 i}{6 i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{2}{7 i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{3}{10 i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{2+6 i}{1+7 i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{5+i}{2+9 i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{3+6 i}{4-5 i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{7-3 i}{4-3 i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{-2+7 i}{-1+i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{-3+8 i}{-2+i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{-1-3 i}{-2-10 i} $$

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{-3-4 i}{-4-11 i} $$

Some of the solution sets for quadratic equations in the next sections in this chapter will contain complex numbers such as \((-4+\sqrt{-12}) / 2\) and \((-4-\sqrt{-12}) / 2\). We can simplify the first number as follows. $$ \begin{aligned} \frac{-4+\sqrt{-12}}{2} &=\frac{-4+i \sqrt{12}}{2} \\ &=\frac{-4+2 i \sqrt{3}}{2}=\frac{2(-2+i \sqrt{3})}{2} \\ &=-2+i \sqrt{3} \end{aligned} $$ Simplify each of the following complex numbers. (Objective 3) (a) \(\frac{-4-\sqrt{-12}}{2}\) (b) \(\frac{6+\sqrt{-24}}{4}\) (c) \(\frac{-1-\sqrt{-18}}{2}\) (d) \(\frac{-6+\sqrt{-27}}{3}\) (e) \(\frac{10+\sqrt{-45}}{4}\) (f) \(\frac{4-\sqrt{-48}}{2}\)

Why is the set of real numbers a subset of the set of complex numbers?

Can the sum of two nonreal complex numbers be a real number? Defend your answer.

Can the product of two nonreal complex numbers be a real number? Defend your answer.