Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
Answer:
(a) y = [tex]\frac{-1}{2}[/tex]x - [tex]\frac{3}{2}[/tex]
(b) y = 2x + 3
Step-by-step explanation:
(a) The equation of a line given by points M(x₁, y₁) and N(x₂, y₂) is given by:
y - y₁ = m(x - x₁) -------------------(i)
Where;
m = [tex]\frac{y_2 - y_1}{x_2 - x_1}[/tex] = slope or gradient of the line ---------------(ii)
Given points on the triangle are:
A(2,5)
B(1,-2)
C(-5,1)
To find the equation of line BC, we use the formulas in equations (i) and (ii) where the points of the line are B(1,-2) and C(-5,1) and;
x₁ = 1
y₁ = -2
x₂ = -5
y₂ = 1
==> First get the gradient using equation (ii) as follows;
m = [tex]\frac{y_2 - y_1}{x_2 - x_1}[/tex] = [tex]\frac{1 - (-2)}{-5 -1}[/tex]
m = [tex]\frac{3}{-6}[/tex]
m = [tex]\frac{-1}{2}[/tex]
==> Now, use equation (i) to find the equation of the line as follows;
y - (-2) = [tex]\frac{-1}{2}[/tex](x - 1)
y + 2 = [tex]\frac{-1}{2}[/tex](x - 1)
Multiply both sides by 2
2(y+2) = -1 ( x - 1 )
2y + 4 = -x + 1
2y = -x + 1 - 4
2y = - x - 3
y = [tex]\frac{-1}{2}[/tex]x - [tex]\frac{3}{2}[/tex]
Therefore, the equation of the line is y = [tex]\frac{-1}{2}[/tex]x - [tex]\frac{3}{2}[/tex]
(b) To find the perpendicular line from A to BC, note that
i. two lines are perpendicular if they meet at 90°
ii. the general equation of a line could also be written as y = mx + c where m is the slope and c is the intercept.
iii. when one line has a slope of m, then a perpendicular line to that line will have a slope of [tex]\frac{-1}{m}[/tex]
The equation of line BC is y = [tex]\frac{-1}{2}[/tex]x - [tex]\frac{3}{2}[/tex].
This means that BC has a slope of [tex]\frac{-1}{2}[/tex]
A perpendicular line from A to BC will have a slope of 2.
Now to get the equation of this perpendicular line from A(2, 5) to BC, we use the general equation of a line given in equation (i)
where;
m = 2
x₁ = 2
y₁ = 5
Substitute these values into equation (i)
y - 5 = 2(x - 2)
Solving by simplification gives;
y - 5 = 2x - 2
y = 2x + 3
Therefore, the equation of the perpendicular line is y = 2x + 3
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.