The distance between the points `(3,pi/4)` and `(7,(5pi)/4)`

Options

A. 8
B. 10
C. 12
D. 14

Answer

To find the distance between the points \( (3, \frac{\pi}{4}) \) and \( (7, \frac{5\pi}{4}) \) given in polar coordinates, we will first convert these points into Cartesian coordinates and then use the distance formula.

### Step 1: Convert the first point \( (3, \frac{\pi}{4}) \) to Cartesian coordinates.
The Cartesian coordinates \( (x, y) \) can be calculated using the formulas:
\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]

For the point \( (3, \frac{\pi}{4}) \):
- \( r = 3 \)
- \( \theta = \frac{\pi}{4} \)

Calculating \( x \) and \( y \):
\[ x_1 = 3 \cos\left(\frac{\pi}{4}\right) = 3 \cdot \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}} \]
\[ y_1 = 3 \sin\left(\frac{\pi}{4}\right) = 3 \cdot \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}} \]

Thus, the Cartesian coordinates of the first point are:
\[ \left(\frac{3}{\sqrt{2}}, \frac{3}{\sqrt{2}}\right) \]

### Step 2: Convert the second point \( (7, \frac{5\pi}{4}) \) to Cartesian coordinates.
For the point \( (7, \frac{5\pi}{4}) \):
- \( r = 7 \)
- \( \theta = \frac{5\pi}{4} \)

Calculating \( x \) and \( y \):
\[ x_2 = 7 \cos\left(\frac{5\pi}{4}\right) = 7 \cdot \left(-\frac{1}{\sqrt{2}}\right) = -\frac{7}{\sqrt{2}} \]
\[ y_2 = 7 \sin\left(\frac{5\pi}{4}\right) = 7 \cdot \left(-\frac{1}{\sqrt{2}}\right) = -\frac{7}{\sqrt{2}} \]

Thus, the Cartesian coordinates of the second point are:
\[ \left(-\frac{7}{\sqrt{2}}, -\frac{7}{\sqrt{2}}\right) \]

### Step 3: Use the distance formula to find the distance between the two points.
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates:
\[ d = \sqrt{\left(-\frac{7}{\sqrt{2}} - \frac{3}{\sqrt{2}}\right)^2 + \left(-\frac{7}{\sqrt{2}} - \frac{3}{\sqrt{2}}\right)^2} \]
\[ = \sqrt{\left(-\frac{10}{\sqrt{2}}\right)^2 + \left(-\frac{10}{\sqrt{2}}\right)^2} \]
\[ = \sqrt{2 \left(-\frac{10}{\sqrt{2}}\right)^2} \]
\[ = \sqrt{2 \cdot \frac{100}{2}} = \sqrt{100} = 10 \]

### Final Answer:
The distance between the points \( (3, \frac{\pi}{4}) \) and \( (7, \frac{5\pi}{4}) \) is \( 10 \) units.