91Ó°ÊÓ

Suppose that \((a, b)\) is on the circle \(x^{2}+y^{2}=r^{2}\). Show that the line \(a x+b y=r^{2}\) is tangent to the circle at \((a, b)\).

The number \(\frac{1}{2}(a+b)\) is called the average, or arithmetic mean, of \(a\) and \(b\). Show that the arithmetic mean of two numbers is between the two numbers; that is, prove that $$ a

Which of the following are true? Unless it is stated otherwise, assume that \(x, y\), and \(\varepsilon\) are real numbers. (a) For every \(x, x0\), there exists a \(y\) such that \(y>\frac{1}{x}\). (d) For every positive \(x\), there exists a natural number \(n\) such that \(\frac{1}{n}

The number \(\sqrt{a b}\) is called the geometric mean of two positive numbers \(a\) and \(b\). Prove that $$ 0

. Prove the following statements. (a) If \(n\) is odd, then \(n^{2}\) is odd. (Hint: If \(n\) is odd, then there exists an integer \(k\) such that \(n=2 k+1 .\) ) (b) If \(n^{2}\) is odd, then \(n\) is odd. (Hint: Prove the contrapositive.)

Prove that \(n\) is odd if and only if \(n^{2}\) is odd. (See Problem \(73 .\) )

For two positive numbers \(a\) and \(b\), prove that $$ \sqrt{a b} \leq \frac{1}{2}(a+b) $$

Express the perpendicular distance between the parallel lines \(y=m x+b\) and \(y=m x+B\) in terms of \(m, b\), and \(B .\) Hint : The required distance is the same as that between \(y=m x\) and \(y=m x+B-b\)

According to the Fundamental Theorem of Arithmetic, every natural number greater than 1 can be written as the product of primes in a unique way, except for the order of the factors. For example, \(45=3 \cdot 3 \cdot 5 .\) Write each of the following as a product of primes. (a) 243 (b) 124 (c) 5100

Show that the line through the midpoints of two sides of a triangle is parallel to the third side. Hint: You may assume that the triangle has vertices at \((0,0),(a, 0)\), and \((b, c)\).