Consider line A which is defined by the equation:y=5/6x-5/2
and the point P(-3,6) and then answer the following questions:
a. How would you find the line (B) that passes through point P and is perpendicular to line A? What is the equation of that line?
b. How would you find the length of the segment of line B from point P to line A?
c. How would you find the midpoint between point P and the intersection of line A and line B ?
Answers
Answer:
- y = -6/5x +12/5
- distance from P to A: (66√61)/61 ≈ 8.4504
- midpoint: (-18/61, 168/61) ≈ (-0.2951, 2.7541)
Step-by-step explanation:
a. The slope of the perpendicular line is the negative reciprocal of the slope of the given line, so is ...
m = -1/(5/6) = -6/5
Then the point-slope form of the desired line through (-3, 6) can be written as ...
y = m(x -h) +k . . . . . line with slope m through (h, k)
y = (-6/5)(x +3) +6
y = -6/5x +12/5 . . . equation of line B
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b. The distance from point P to the intersection point (X) can be found from the formula for the distance from a point to a line.
When the line's equation is written in general form, ax+by+c=0, the distance from point (x, y) to the line is ...
d = |ax +by +c|/√(a² +b²)
The equation of line A can be written in general form as ...
y = 5/6x -5/2
6y = 5x -15
5x -6y -15 = 0
Then the distance from P to the line is ...
d = |5(-3) -6(6) -15|/√(5² +(-6)²) = 66/√61
The length of segment PX is (66√61)/61.
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c. To find the midpoint, we need to know the point of intersection, X. We find that by solving the simultaneous equations ...
y = 5/6x -5/2
y = -6/5x +12/5
Equating y-values gives ...
5/6x -5/2 = -6/5x +12/5
Adding 6/5x +5/2 gives ...
x(5/6+6/5) = 12/5 +5/2
x(61/30) = 49/10
x = (49/10)(30/61) = 147/61
y = 5/6(147/61) -5/2 = -30/61
Then the point of intersection of the lines is X = (147/61, -30/61).
So, the midpoint of PX is ...
M = (P +X)/2
M = ((-3, 6) +(147/61, -30/61))/2
M = (-18/61, 168/61)
Final answer:
To find line B perpendicular to line A and pass through point P, calculate the negative reciprocal of line A's slope and use it in the line equation along with point P coordinates to find c. The segment length from point P to line A is calculated using the distance formula and involves finding the intersection point between lines A and B. The midpoint is calculated using the midpoint formula.
Explanation:
To answer this question, we need to understand that two lines are perpendicular if the product of their slopes is -1. Line A has a slope of 5/6. Therefore, the slope of line B, perpendicular to line A, is -6/5 (the negative reciprocal). The equation of a line is y = mx + c where m is the slope and c is the y-intercept. As line B passes through point P(-3,6), we can substitute these values into the line equation y = -6/5x + c to solve for c. This will give us the equation of line B.
To find the length of the segment from point P to Line A, we would first need to find the intersection point of Line A and B. Then use the distance formula, which is sqrt[(x2-x1)^2 + (y2-y1)^2].
The midpoint of two points, (x1,y1) and (x2,y2) is given by ((x1+x2)/2, (y1+y2)/2). This formula can be used to find the midpoint between point P and the intersection of line A and line B.
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What does the equation look like now?
Use your answer and finish solving the equation for r.
r=
Answers
Answer:
r=6
Step-by-step explanation:
subtract r from 5r to get 4r, and add 9 to the other side, 4r=24, divide 24 by 4 to get 6
4 units
B.
5 units
C.
8 units
D.
9 units
Answers
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d²=(3-8)²+(1-1)²=25+0=25
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2x-5y=7
3x-2y=-17
Answers
Answers
Answer:
C''(-2, -2)
Step-by-step explanation:
Reflection over the x-axis negates the y-coordinate. Reflection over the y-axis negates the x-coordinate. Negating both coordinates of C(2, 2) makes it ...
C''(-2, -2)
if the employee earned $570 in a week, what was the amount of her sales for that week?
Answers
Answer:
Weekly earning of employee, E(w) is given by 450+0.1w, where w is the weekly sales.
Weekly sales for that week = 1200 $
Step-by-step explanation:
A phone store employee earns a salary of $450 per week plus 10% commission on her weekly sales.
Let w be the weekly sales, the weekly earning of employee, E(w) is given by
So
Weekly earning of employee, E(w) is given by 450+0.1w, where w is the weekly sales.
Now we need to find what was the amount of her sales for that week, if she earned 570 $
That is
570 = 450+0.1w
0.1 w = 570 - 450 = 120
w = 1200 $
Weekly sales = 1200 $