Question

Given a line whose equation is \(y=-\frac{1}{4} x+7\) which line below definitely forms part of a right triangle with the given line? A. \(y=\frac{1}{4} x+9\) B. \(y=-4 x+7\) C. \(y=-\frac{1}{4} x+5\) D. \(y=4 x+11\)

Short answer

Option D: \( y = 4 x + 11 \) forms a right triangle with the given line.

Step-by-step solution

Identify the slope of the given line

The given line equation is \( y = -\frac{1}{4} x + 7 \). The slope of this line is the coefficient of \( x \), which is \( -\frac{1}{4} \).

Determine the condition for perpendicular lines

Two lines are perpendicular if the product of their slopes is \( -1 \). Therefore, if the slope of the given line is \( m_1 = -\frac{1}{4} \), then for a line to be perpendicular to it, its slope \( m_2 \) must satisfy \( m_1 \times m_2 = -1 \).

Calculate the perpendicular slope

Given \( m_1 = -\frac{1}{4} \), solve for \( m_2 \) using the equation \( m_1 \times m_2 = -1 \): \[ -\frac{1}{4} \times m_2 = -1 \] \[ m_2 = 4 \] This means the slope of the line that is perpendicular to the given line must be \( 4 \).

Identify the line with slope 4

Inspect the options to find the line with slope \( m_2 = 4 \): A. \( y = \frac{1}{4} x + 9 \): Slope = \( \frac{1}{4} \) B. \( y = -4 x + 7 \): Slope = \( -4 \) C. \( y = -\frac{1}{4} x + 5 \): Slope = \( -\frac{1}{4} \) D. \( y = 4 x + 11 \): Slope = \( 4 \)

Conclusion

The line \( y = 4 x + 11 \) (Option D) has a slope of \( 4 \) and will form a right triangle with the given line.

Key concepts

linear equations

In algebra, a linear equation is an equation which forms a straight line when graphed. The standard form of a linear equation is written as \( y = mx + b \). Here, \( m \) represents the slope (steepness) of the line, and \( b \) is the y-intercept (the point where the line crosses the y-axis).
Linear equations are foundational in understanding many algebraic concepts and are used frequently to depict relationships between variable quantities.
In our exercise, the line given by the equation \( y = -\frac{1}{4} x + 7 \) is a perfect example of a linear equation.
Knowing how to identify and manipulate these equations can be crucial in solving geometry and algebra problems.

slope

The slope of a line is a number that describes both the direction and the steepness of the line. It is often denoted by the letter \( m \) in the slope-intercept form (\( y = mx + b \)).
The slope is defined as the ratio of the vertical change to the horizontal change between two points on a line:
\frac{rise}{run}\br> In the exercise, the slope of the given line is \( -\frac{1}{4} \), which tells us that for every 4 units the line moves horizontally, it moves down 1 unit vertically.
Understanding slopes is very crucial. They can reveal parallelism (same slope) or perpendicularity (product of slopes is -1) between two lines, influencing their geometric relationships.

perpendicular lines

Perpendicular lines are lines that intersect at a right angle (90 degrees).
One key property to identify perpendicular lines is that the product of their slopes is \( -1 \).
For instance, in our exercise, the slope of the given line is \( -\frac{1}{4} \). To find a perpendicular line, we calculate the slope that, when multiplied by \( -\frac{1}{4} \), equals \(-1 \). This turns out to be \( 4 \).
Hence, any line with a slope of \( 4 \) will be perpendicular to the given line \( y = - \frac{1}{4} x + 7 \).
Perpendicular lines are significant as they help to form right angles, which are fundamental in many areas of geometry and algebra.

geometric relationships

Geometric relationships involve understanding how different shapes, lines, and angles interact and coexist in space.
In the context of algebra, understanding the relationships between lines helps in gaining insights into their interactions.
Right triangles, for instance, are formed when one line is perpendicular to another, creating a 90-degree angle.
In the given problem, recognizing that line \( y = 4 x + 11 \) is perpendicular to \( y = -\frac{1}{4} x + 7 \) allows us to determine that these lines can indeed form a right triangle.
Realizing these geometric relationships can simplify complex problems by breaking them into more manageable parts, making them easier to solve.