For the line joining points A(-1, 3) and B(5, 7):
a.Find the midpoint of the line.
b.Find the distance of the line.
c.Find the slope of the line.
d.Find the y-intercept of the line.
e.What would be the slope of a line perpendicular to this line?
GIVEN: A(-1,3), B((5,7).
a. X = (-1+5)/ 2 = 2.
Y = (3 + 7)/2 = 5.
Mid-point: (2,5).
b. d^2 = (5 + 1)^2 +(7-3)^2 = 36 + 16
= 52 d = sqrt (52) = 7.2.
c. Slope = (7-3) / (5 + 1) = 2/3.
d. Y = mX + b
3 = 2/3(-1) + b, b = 3 2/3 = 11/3 =
y - int.
e. m2 = -3/2 = neg. reciprocal of m1.
a. To find the midpoint of the line joining two points, you can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint are equal to the average of the coordinates of the two points.
For points A(-1, 3) and B(5, 7), the formula becomes:
Midpoint = ( (x1 + x2) / 2 , (y1 + y2) / 2 )
Substituting the coordinates of A and B into the formula:
Midpoint = ( (-1 + 5) / 2 , (3 + 7) / 2 )
= (4 / 2, 10 / 2)
= (2, 5)
Therefore, the midpoint of the line joining points A(-1, 3) and B(5, 7) is (2, 5).
b. To find the distance of the line joining two points, you can use the distance formula. The distance formula calculates the length of a line segment between two points in a coordinate system.
The distance formula is given by:
Distance = sqrt( (x2 - x1)^2 + (y2 - y1)^2 )
Substituting the coordinates of A and B into the formula:
Distance = sqrt( (5 - (-1))^2 + (7 - 3)^2 )
= sqrt( (6)^2 + (4)^2 )
= sqrt( 36 + 16 )
= sqrt( 52 )
= 2 * sqrt(13)
Therefore, the distance of the line joining points A(-1, 3) and B(5, 7) is 2 * sqrt(13).
c. To find the slope of the line, you can use the slope formula. The slope between two points is calculated as the change in y-coordinates divided by the change in x-coordinates.
The slope formula is given by:
Slope = (y2 - y1) / (x2 - x1)
Substituting the coordinates of A and B into the formula:
Slope = (7 - 3) / (5 - (-1))
= 4 / 6
= 2 / 3
Therefore, the slope of the line joining points A(-1, 3) and B(5, 7) is 2 / 3.
d. The equation of a line in the form y = mx + b, where m is the slope and b is the y-intercept. To find the y-intercept of the line, you need the equation of the line.
Using the point-slope form:
y - y1 = m(x - x1)
Substituting the coordinates of A and the slope from part (c):
y - 3 = (2/3)(x - (-1))
y - 3 = (2/3)(x + 1)
y - 3 = (2/3)x + 2/3
y = (2/3)x + 2/3 + 3
y = (2/3)x + 9/3
y = (2/3)x + 3
Comparing this equation with y = mx + b, we can see that the y-intercept of the line joining points A(-1, 3) and B(5, 7) is 3.
e. The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line. The negative reciprocal is found by taking the opposite sign of the slope and flipping the fraction.
The slope of the given line is 2/3, so the negative reciprocal would be:
Negative Reciprocal = -(1 / (2/3))
= -(3/2)
= -3/2
Therefore, the slope of a line perpendicular to the line joining points A(-1, 3) and B(5, 7) is -3/2.