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Sagot :
Answer:
y=3x-21
Step-by-step explanation:
General outline
- Find equation for line AB
- Find equation for perpendicular line BC
Step 1. Find equation for line AB
Given points A(2,5) and B(8,3), line AB must contain them.
To calculate the slope, [tex]m_{\text{AB}}[/tex], of line AB, use the slope formula:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m_{\text{AB}}=\dfrac{(3)-(5)}{(8)-(2)}[/tex]
[tex]m_{\text{AB}}=\dfrac{-2}{6}[/tex]
[tex]m_{\text{AB}}=-\frac{1}{3}[/tex]
Since the slope isn't undefined, line AB must cross the y-axis somewhere. To find the y-intercept, build and equation in slope-intercept form:
[tex]y=m_{\text{AB}}x+b_{\text{AB}}[/tex]
[tex]y=\left(-\frac{1}{3} \right) x+b_{\text{AB}}[/tex]
Substituting values for a known point (point A) on line AB...
[tex](5)=\left(-\frac{1}{3} \right) (2)+b_{\text{AB}}[/tex]
[tex]5=-\frac{2}{3} +b_{\text{AB}}[/tex]
[tex](5)+\frac{2}{3} =(-\frac{2}{3} +b_{\text{AB}})+\frac{2}{3}[/tex]
Finding a common denominator...
[tex]\frac{3}{3}*5+\frac{2}{3} =b_{\text{AB}}[/tex]
[tex]\frac{15}{3}+\frac{2}{3} =b_{\text{AB}}[/tex]
[tex]\frac{17}{3}=b_{\text{AB}}[/tex]
So, the equation for line AB is [tex]y=-\frac{1}{3} x +\frac{17}{3}[/tex]
Step 2. Find equation for line BC
Since line AB and line BC form a right angle, they are perpendicular. Perpendicular lines have slopes that are opposite (opposite sign) reciprocals (fraction flipped upside-down) of each other. Stated another way, the slopes multiply to make negative 1.
[tex]m_{\text{AB}}*m_{\text{BC}}=-1[/tex]
[tex]\left( -\frac{1}{3} \right) *m_{\text{BC}}=-1[/tex]
[tex]-3*\left( -\frac{1}{3} *m_{\text{BC}} \right) =-3*(-1)[/tex]
[tex]m_{\text{BC}} =3[/tex]
Since the slope isn't undefined, line BC must also cross the y-axis somewhere. To find the y-intercept, build and equation in slope-intercept form:
[tex]y=m_{\text{BC}}x+b_{\text{BC}}[/tex]
[tex]y=(3) x+b_{\text{BC}}[/tex]
Substituting values for a known point (point B) on line BC...
[tex](3)=3 * (8)+b_{\text{BC}}[/tex]
[tex]3=24+b_{\text{BC}}[/tex]
[tex](3)-24=(24+b_{\text{BC}})-24[/tex]
[tex]-21=b_{\text{BC}}[/tex]
So, the equation for line BC is [tex]y=3 x -21[/tex]
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