Question

Find the third proportional of (3 + √2) and 2√7.

• A. 2(3+√2)
• B. (12-√8)
• C. 4(3-√2)
• D. (12+√8)

Answer 1

Answer:

C. 4(3-√2)

The third proportional (c) to two numbers (a) and (b) is defined by:
$a : b = b : c \quad \Rightarrow \quad c = \frac{b^2}{a}$
$a = (3 + \sqrt{2}), \quad b = 2\sqrt{7}$
Square (b)

$b^2 = (2\sqrt{7})^2 = 4 \times 7 = 28$

Find the third proportional

$c = \frac{28}{3 + \sqrt{2}}$

Rationalize the denominator:
$c = \frac{28(3 - \sqrt{2})}{(3 + \sqrt{2})(3 - \sqrt{2})}$

$= \frac{28(3 - \sqrt{2})}{9 - 2}$

$= \frac{28(3 - \sqrt{2})}{7}$

$c = 4(3 - \sqrt{2}) = 12 - 4\sqrt{2}=4(3-\sqrt2)$