Consider line CD passing through points [tex]$C (-5,10)$[/tex] and [tex]$D (1,8)$[/tex]. What is the slope of [tex]$\overrightarrow{C D}$[/tex]?
Answer (2)
Identify the coordinates of points C and D: C ( − 5 , 10 ) and D ( 1 , 8 ) .
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates into the formula: m = 1 − ( − 5 ) 8 − 10 .
Simplify to find the slope: m = − 3 1 .
Explanation
Identify coordinates and state the slope formula. First, let's identify the coordinates of the two points that the line CD passes through. We have point C at (-5, 10) and point D at (1, 8). The slope of a line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula:
m = x 2 − x 1 y 2 − y 1
where m represents the slope of the line.
Substitute coordinates into the formula. Now, let's substitute the coordinates of points C and D into the slope formula. We have:
x 1 = − 5 , y 1 = 10 x 2 = 1 , y 2 = 8
Plugging these values into the formula, we get:
m = 1 − ( − 5 ) 8 − 10
Simplify the expression to find the slope. Next, we simplify the expression:
m = 1 + 5 − 2 m = 6 − 2 m = − 3 1
So, the slope of line CD is − 3 1 .
State the final answer. Therefore, the slope of line CD is − 3 1 .
Examples
Understanding slope is crucial in many real-world applications. For example, when designing roads or ramps, engineers need to calculate the slope to ensure they are safe and accessible. A steeper slope requires more effort to climb, whether it's a car driving up a hill or a person using a wheelchair on a ramp. By calculating the slope, engineers can optimize the design for efficiency and safety. Similarly, in construction, the slope of a roof is important for water runoff and preventing leaks. A well-calculated slope ensures that rainwater doesn't accumulate on the roof, which could cause damage over time.
The slope of line CD, which passes through points C (-5, 10) and D (1, 8), is -\frac{1}{3}. This negative slope indicates a downward trend from point C to point D as x increases.
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