The line Q passes through the points (-10,-2) and (-8,-8).
The line R passes through the points (1,2) and (10,a).
The lines Q and R are perpendicular.
Find a.
Thanks.
slope of line Q = (-8+2)/(-8+10) = -6/2 = -3
slope of line R = (a-2)/(10-1) = (a-2)/9
but they are perpendicular, so
(a-2)/9 = -1/3
continue
Well, to find the value of "a" we need to determine the slope of line R first.
Let's start by finding the slope (m1) of line Q using the formula:
m1 = (y2 - y1) / (x2 - x1)
Substituting the coordinates (-10,-2) and (-8,-8) into the slope formula, we get:
m1 = (-8 - (-2)) / (-8 - (-10))
= (-8 + 2) / (-8 + 10)
= -6 / 2
= -3
Since Q and R are perpendicular, the product of their slopes should be -1:
m1 * m2 = -1
Substituting the known value of m1 as -3, we have:
-3 * m2 = -1
Now, we can solve for m2:
m2 = -1 / -3
= 1/3
So, the slope of line R (m2) is 1/3.
Now let's find the value of "a" using the slope of R and the point (1,2) on the line:
m2 = (y2 - y1) / (x2 - x1)
Substituting the coordinates (1, 2) and (10, a) into the slope formula, we have:
1/3 = (a - 2) / (10 - 1)
1/3 = (a - 2) / 9
To solve for a, we can cross-multiply:
9 = 3(a - 2)
9 = 3a - 6
3a = 15
a = 5
So, the value of "a" is 5.
Hope that answers your question!
To determine the value of a, we can use the concept of the slope of perpendicular lines.
The slope (m) of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
For line Q:
Point 1: (-10, -2)
Point 2: (-8, -8)
mQ = (-8 - (-2)) / (-8 - (-10))
= (-8 + 2) / (-8 + 10)
= -6 / 2
= -3
For line R:
Point 1: (1, 2)
Point 2: (10, a)
mR = (a - 2) / (10 - 1)
Since lines Q and R are perpendicular, their slopes are negative reciprocals of each other.
Therefore, mQ * mR = -1:
-3 * ((a - 2) / (10 - 1)) = -1
Multiplying both sides by (10 - 1):
-30(a - 2) = -10
Expanding:
-30a + 60 = -10
Subtracting 60 from both sides:
-30a = -10 - 60
-30a = -70
Dividing by -30:
a = -70 / -30
a = 7/3
Therefore, a is equal to 7/3.
To find the value of "a" for the line R, we need to use the information that the lines Q and R are perpendicular.
First, let's find the slope of the line Q using the formula:
slope = (change in y-coordinates) / (change in x-coordinates).
For the points (-10, -2) and (-8, -8):
slope of Q = (-8 - (-2)) / (-8 - (-10))
= -6 / 2
= -3
Since a line perpendicular to another line has a slope that is the negative reciprocal of the original line, we can determine the slope of line R. The negative reciprocal of -3 is 1/3.
Now, with the given points (1, 2) and (10, a) for line R, we can find the slope of R using the same formula:
slope of R = (change in y-coordinates) / (change in x-coordinates)
= (a - 2) / (10 - 1)
= (a - 2) / 9
Since line R is perpendicular to line Q, the slopes of Q and R are negative reciprocals. Therefore, we can equate the two slopes and solve for "a":
-3 = 1/3
Cross-multiplying:
-3 * 9 = 1 * (a - 2)
-27 = a - 2
a = -27 + 2
a = -25
Therefore, the value of "a" for the line R is -25.