Line $AB$ passes through $A(-3,0)$ and $B(-6,5)$. What is the equation of the line that passes through the origin and is parallel to line $AB$?
A. $5x-3y=0$
B. $-x+3y=0$
C. $-5x-3y=0$
D. $3x+5y=0$
E. $-3x+5y=0
Answer (2)
Calculate the slope of line A B using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 3 5 .
Since the required line is parallel to line A B , it has the same slope, m = − 3 5 .
Use the slope-intercept form of a line, y = m x , and substitute the slope to get y = − 3 5 x .
Rewrite the equation in the standard form a x + b y = 0 , which gives − 5 x − 3 y = 0 .
Explanation
Problem Analysis We are given two points A ( − 3 , 0 ) and B ( − 6 , 5 ) on line A B . We need to find the equation of a line that passes through the origin and is parallel to line A B .
Calculate the slope of line AB First, we need to find the slope of line A B . The slope m is given by the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of points A and B , respectively. Substituting the coordinates of A ( − 3 , 0 ) and B ( − 6 , 5 ) into the formula, we get: m = − 6 − ( − 3 ) 5 − 0 = − 6 + 3 5 = − 3 5 = − 3 5
Determine the slope of the parallel line Since the line we are looking for is parallel to line A B , it will have the same slope as line A B . Therefore, the slope of the parallel line is also m = − 3 5 .
Write the equation of the parallel line The equation of a line passing through the origin ( 0 , 0 ) with slope m is given by: y = m x Substituting m = − 3 5 into the equation, we get: y = − 3 5 x
Rewrite the equation in standard form To rewrite the equation in the standard form a x + b y = 0 , we can multiply both sides of the equation by 3 to eliminate the fraction: 3 y = − 5 x Now, add 5 x to both sides of the equation: 5 x + 3 y = 0 Multiplying by − 1 , we get: − 5 x − 3 y = 0
Final Answer The equation of the line that passes through the origin and is parallel to line A B is − 5 x − 3 y = 0 .
Examples
Understanding parallel lines is crucial in architecture. When designing buildings, architects use parallel lines to create symmetrical and balanced structures. For instance, the walls of a room are often parallel to each other, ensuring stability and visual harmony. The concept of slope is also vital, as it helps determine the steepness of roofs and ramps, ensuring proper drainage and accessibility. By applying these mathematical principles, architects can create functional and aesthetically pleasing designs.
The equation of the line that passes through the origin and is parallel to line AB is − 5 x − 3 y = 0 . This was determined by first calculating the slope of line AB and using that slope to write the equation of the parallel line. The answer corresponds to option C.
;
