What is the distance between the x-intercept and the y-intercept of the line `y=(2)/(3)x-6`?

Options

A. `9`
B. `15`
C. `sqrt(89)`
D. `sqrt(117)`

Answer

To find the distance between the x-intercept and the y-intercept of the line given by the equation \( y = \frac{2}{3}x - 6 \), we will follow these steps:

### Step 1: Find the x-intercept
The x-intercept occurs when \( y = 0 \). We set the equation equal to zero and solve for \( x \):

\[ 0 = \frac{2}{3}x - 6 \]

Adding 6 to both sides:

\[ \frac{2}{3}x = 6 \]

Now, multiply both sides by \(\frac{3}{2}\) to isolate \( x \):

\[ x = 6 \times \frac{3}{2} = 9 \]

Thus, the x-intercept is \( (9, 0) \).

### Step 2: Find the y-intercept
The y-intercept occurs when \( x = 0 \). We substitute \( x = 0 \) into the equation:

\[ y = \frac{2}{3}(0) - 6 = -6 \]

Thus, the y-intercept is \( (0, -6) \).

### Step 3: Use the distance formula
Now, we need to find the distance between the two points \( (9, 0) \) and \( (0, -6) \). The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of the x-intercept and y-intercept:

\[ d = \sqrt{(0 - 9)^2 + (-6 - 0)^2} \]

Calculating the squares:

\[ d = \sqrt{(-9)^2 + (-6)^2} = \sqrt{81 + 36} \]

Adding the squares:

\[ d = \sqrt{117} \]

### Step 4: Simplify the distance
The distance can be left as \( \sqrt{117} \), which is approximately \( 10.82 \) when calculated.

Thus, the distance between the x-intercept and the y-intercept of the line is \( \sqrt{117} \).

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