Select the correct answer.
$\overleftrightarrow{A B}$ and $\overleftrightarrow{B C}$ form a right angle at point $B$. If $A=(-3,-1)$ and $B=(4,4)$, what is the equation of $\overleftrightarrow{B C}$ ?
A. $x+3 y=16$
B. $2 x+y=12$
C. $-7 x-5 y=-48$
D. $7 x-5 y=48$
Answer (2)
Calculate the slope of line A B using points A ( − 3 , − 1 ) and B ( 4 , 4 ) : m A B = 7 5 .
Determine the slope of line BC , which is perpendicular to A B : m BC = − 5 7 .
Use the point-slope form with point B ( 4 , 4 ) and slope m BC to find the equation: y − 4 = − 5 7 ( x − 4 ) .
Convert the equation to standard form: 7 x + 5 y = 48 .
Explanation
Problem Analysis The lines A B and BC form a right angle at point B . This means that the lines are perpendicular. We are given the coordinates of points A = ( − 3 , − 1 ) and B = ( 4 , 4 ) . Our goal is to find the equation of the line BC .
Calculate the slope of AB First, we need to find the slope of the line A B . The slope m 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 Using the coordinates of points A = ( − 3 , − 1 ) and B = ( 4 , 4 ) , we have: m A B = 4 − ( − 3 ) 4 − ( − 1 ) = 7 5 So, the slope of line A B is 7 5 .
Calculate the slope of BC Since A B and BC are perpendicular, the slope of BC is the negative reciprocal of the slope of A B . Therefore: m BC = − m A B 1 = − 7 5 1 = − 5 7 = − 1.4 So, the slope of line BC is − 5 7 .
Find the equation of BC Now we use the point-slope form of a line to find the equation of BC . The point-slope form is given by: y − y 1 = m ( x − x 1 ) where ( x 1 , y 1 ) is a point on the line and m is the slope. We know that point B = ( 4 , 4 ) lies on BC , and m BC = − 5 7 . Plugging these values into the point-slope form, we get: y − 4 = − 5 7 ( x − 4 ) To convert this to standard form A x + B y = C , we first multiply both sides by 5 to eliminate the fraction: 5 ( y − 4 ) = − 7 ( x − 4 ) 5 y − 20 = − 7 x + 28 Now, rearrange the equation to get the standard form: 7 x + 5 y = 28 + 20 7 x + 5 y = 48
Final Answer The equation of line BC is 7 x + 5 y = 48 . Comparing this with the given options, we see that option D matches our result.
Conclusion Therefore, the correct answer is D. 7 x + 5 y = 48 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For example, when building a house, the walls need to be perpendicular to the ground to ensure stability. If you know the slope of the ground (line AB), you can calculate the slope of the wall (line BC) to make sure it's perfectly upright. This ensures the structure is safe and sound.
The equation of the line BC that is perpendicular to line A B at point B is 7 x + 5 y = 48 , which corresponds to option D. After calculating the slopes and using the point-slope form, we derived the standard form equation. Thus, the correct answer is D.
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