Question

The line \(y = 5x + 7\) is perpendicular to the line joining the points \((2, 12)\) and \((12, k)\). Then the value of \(k\) is equal to:

• A. 12
• B. -12
• C. 8
• D. -8
• E. 10

Hint:

In problems involving perpendicular lines, ensure that the product of their slopes equals \(-1\). This is a fundamental property of perpendicular lines in a coordinate plane.

Answer 1

Answer: (E)

The slope of the line \(y = 5x + 7\) is \(5\). For two lines to be perpendicular, the product of their slopes must be \(-1\). 
Thus, we need to find the slope of the line joining \((2, 12)\) and \((12, k)\). Calculate the slope of this line: \[ {slope} = \frac{k - 12}{12 - 2} = \frac{k - 12}{10} \] Set the product of the slopes to \(-1\): \[ 5 \cdot \frac{k - 12}{10} = -1 \] \[ k - 12 = -2 \times 10 = -20 \] \[ k = -20 + 12 = -8 \] However, to verify against the correct answer provided, let's recheck the calculation: \[ 5 \cdot \frac{k - 12}{10} = -1 \] \[ 5(k - 12) = -10 \] \[ k - 12 = -2 \] \[ k = 10 \] Thus, the correct value for \(k\) that makes the lines perpendicular is \(10\).