distance (1/4,-1/7),(3/4,6/7) — step-by-step solution

The distance is the norm of the difference of the points: the square root of the sum of the squared coordinate differences

1) Let's evaluate:

‖ (\frac{3}{4}, \frac{6}{7}) - (\frac{1}{4}, - \frac{1}{7}) ‖

ⅰ

(\frac{3}{4}, \frac{6}{7}) - (\frac{1}{4}, - \frac{1}{7})

(\frac{3}{4} - \frac{1}{4}, \frac{6}{7} + \frac{1}{7})

‖ (\frac{3}{4} - \frac{1}{4}, \frac{6}{7} + \frac{1}{7}) ‖

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{3}{4} - \frac{1}{4}

\frac{3 - 1}{4}

\frac{6}{7} + \frac{1}{7}

\frac{6 + 1}{7}

‖ (\frac{3 - 1}{4}, \frac{6 + 1}{7}) ‖

ⓘ The numerator and the denominator are computed before the division:

3 - 1

2

6 + 1

7

‖ (\frac{2}{4}, \frac{7}{7}) ‖

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{2}{4}

\frac{1 \times 2}{2 \times 2}

\frac{1}{2}

■We simplify the fraction by dividing the numerator and the denominator by

7

, their greatest common divisor:

\frac{7}{7}

1

‖ (\frac{1}{2}, 1) ‖

ⅰ

‖ (\frac{1}{2}, 1) ‖

\sqrt{{(\frac{1}{2})}^{2} + 1}

\sqrt{{(\frac{1}{2})}^{2} + 1}

ⓘ The power of a quotient is the quotient of the powers, (a/b)^n = a^n/b^n:

{(\frac{1}{2})}^{2}

\frac{1}{2^{2}}

\sqrt{\frac{1}{2^{2}} + 1}

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

2^{2}

2 \times 2

4

\sqrt{\frac{1}{4} + 1}

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{1}{4} + 1

\frac{1 + 4}{4}

\sqrt{\frac{1 + 4}{4}}

ⓘ The numerator and the denominator are computed before the division:

1 + 4

5

\sqrt{\frac{5}{4}}

ⓘ The root of a quotient is the quotient of the roots, sqrt(a/b) = sqrt(a)/sqrt(b):

\sqrt{\frac{5}{4}}

\frac{\sqrt{5}}{\sqrt{4}}

\frac{\sqrt{5}}{\sqrt{4}}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{4}

2

;

2^{2}

4

▶Distance:

\frac{\sqrt{5}}{2}

2) The numeric value of:

\frac{\sqrt{5}}{2}

is:

\frac{\approx 2.2361}{2}

▷

\approx 1.1180