Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar.
$\overleftrightarrow{A B}$ and $\overleftrightarrow{B C}$ form a right angle at their point of intersection, $B$.
If the coordinates of $A$ and $B$ are $(14,-1)$ and $(2,1)$, respectively, the $y$-intercept of $\overleftrightarrow{A B}$ is $\square$ and the equation of $B C$ is $y=\square x+\square$.
If the $y$-coordinate of point $C$ is 13 , its $x$-coordinate is $\square$
Answer (2)
Calculate the slope of line A B using points A ( 14 , − 1 ) and B ( 2 , 1 ) : m A B = − 6 1 .
Determine the y-intercept of line A B using the slope and point B : b = 3 4 .
Find the equation of line BC using the negative reciprocal of m A B and point B : y = 6 x − 11 .
Calculate the x-coordinate of point C by substituting y = 13 into the equation of line BC : x = 4 .
3 4 6 − 11 4
Explanation
Calculate the slope of line AB First, we need to find the slope of line A B using the coordinates of points A ( 14 , − 1 ) and B ( 2 , 1 ) . The slope m A B is calculated as follows: m A B = x 2 − x 1 y 2 − y 1 = 2 − 14 1 − ( − 1 ) = − 12 2 = − 6 1
Determine the y-intercept of line AB Next, we find the y-intercept of line A B . We use the slope-intercept form of a line, y = m A B x + b , and plug in the coordinates of point B ( 2 , 1 ) and the slope m A B = − 6 1 to solve for b :
1 = − 6 1 ( 2 ) + b 1 = − 3 1 + b b = 1 + 3 1 = 3 4 So, the y-intercept of line A B is 3 4 .
Find the equation of line BC Since A B and BC are perpendicular, the slope of line BC is the negative reciprocal of the slope of line A B . Therefore: m BC = − m A B 1 = − − 6 1 1 = 6 Now, we find the equation of line BC using the slope-intercept form, y = m BC x + b , and plug in the coordinates of point B ( 2 , 1 ) and the slope m BC = 6 to solve for b :
1 = 6 ( 2 ) + b 1 = 12 + b b = 1 − 12 = − 11 So, the equation of line BC is y = 6 x − 11 .
Calculate the x-coordinate of point C Finally, we need to find the x-coordinate of point C given that its y-coordinate is 13. We plug y = 13 into the equation of line BC , y = 6 x − 11 , and solve for x :
13 = 6 x − 11 6 x = 13 + 11 6 x = 24 x = 6 24 = 4 So, the x-coordinate of point C is 4.
State the final answer The y-intercept of A B is 3 4 , the equation of BC is y = 6 x − 11 , and the x-coordinate of point C is 4.
Examples
Understanding coordinate geometry is crucial in various real-world applications. For instance, civil engineers use coordinate systems to design roads, bridges, and buildings. Architects rely on coordinate geometry to create precise blueprints and 3D models of structures. Even in video games, developers use coordinate systems to define the positions and movements of characters and objects within the game world. By mastering these concepts, you're building a foundation for success in many technical fields.
The y-intercept of line AB is 4/3, the equation of line BC is y = 6x - 11, and the x-coordinate of point C is 4.
;
