Parallel & Perpendicular Lines Quick Check

Question

5 of 55 of 5 Items
Question
Which line is perpendicular to the line y=5x−2
and goes through the point (2,3)
?(1 point)
Responses

y+3=1/5(x−2)
y plus 3 equals Start Fraction 1 over 5 End Fraction left parenthesis x minus 2 right parenthesis

y−3=5(x−2)
y minus 3 equals 5 left parenthesis x minus 2 right parenthesis

y+3=−1/5(x+2)
y plus 3 equals negative Start Fraction 1 over 5 End Fraction left parenthesis x plus 2 right parenthesis

y−3=−1/5(x−2)

Answer 1

To find the line that is perpendicular to the line \(y = 5x - 2\) and goes through the point \((2, 3)\), we need to follow these steps:

1. Determine the slope of the given line: The given line has a slope of \(5\) (since it is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope).

2. Find the slope of the perpendicular line: The slope of the line perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, if the slope of the original line is \(5\), the slope of the perpendicular line will be \(-\frac{1}{5}\).

3. Use the point-slope form of the line equation: With the slope \(-\frac{1}{5}\) and the point \((2, 3)\), we use the point-slope form \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point and \(m\) is the slope.

Substituting the values we have:

• \(x_1 = 2\)
• \(y_1 = 3\)
• \(m = -\frac{1}{5}\)

This gives us: \[ y - 3 = -\frac{1}{5}(x - 2) \]

Now, let's review the response options given:

1. \(y + 3 = \frac{1}{5}(x - 2)\) -- not correct, slope is wrong.
2. \(y - 3 = 5(x - 2)\) -- not correct, slope is wrong.
3. \(y + 3 = -\frac{1}{5}(x + 2)\) -- not correct, wrong point.
4. \(y - 3 = -\frac{1}{5}(x - 2)\) -- this matches our derived equation.

Thus, the correct answer is: \[ y - 3 = -\frac{1}{5}(x - 2) \]