The coordinates of the end-points of a line segment PQ are P(3,7) and Q(11,-6). Find the coordinates of the point R on the y-axis such that PR = QR.
Please include the workings, as I would like to be able to perform these problems unassisted in the future (I just have trouble remembering the formulas).
Thank you anyone who answers this question :D
A point on the y-axis has coordinates (0,y).
The distance from (0,y) to P(3,7) is the diagonal of a rectangle which has sides (3-0) and (7-y)
similarly for Q: the distances are (11-0) and (-6-y)
So,
PR = √(3^2 + (7-y)^2)
QR = √(11^2 + (-6-y)^2)
to have PR=QR, you need
√(3^2 + (7-y)^2) = √(11^2 + (-6-y)^2)
square both sides to get rid of the radicals:
9+(7-y)^2 = 121+(6+y)^2
9+49-14y+y^2 = 121+36+12y+y^2
collect terms (the y^2's go away - yay!)
26y = 99
y = 99/26
so, the point is (0,99/26)
odd answer, but hey, 99/26 is just a number, like 3 or -5.
oops,
26y = -99, so
y = -99/26
To find the coordinates of point R on the y-axis such that PR = QR, we can use the midpoint formula and the distance formula.
Step 1: Find the midpoint of line segment PQ.
To find the midpoint, we can average the x-coordinates of P and Q and the y-coordinates of P and Q.
Midpoint(x,y) = ((x1 + x2) / 2, (y1 + y2) / 2)
Using the coordinates of P and Q, we get:
Midpoint(x,y) = ((3 + 11) / 2, (7 + -6) / 2)
Midpoint(x,y) = (14/2, 1/2)
Midpoint(x,y) = (7, -2.5)
So, the midpoint of line segment PQ is (7, -2.5).
Step 2: Find the distance between point P and the midpoint.
To find the distance between two points, we can use the distance formula.
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of P (3,7) and the midpoint (7,-2.5), we get:
Distance = sqrt((7 - 3)^2 + (-2.5 - 7)^2)
Distance = sqrt(4^2 + (-9.5)^2)
Distance = sqrt(16 + 90.25)
Distance = sqrt(106.25)
Distance = 10.308
Step 3: Find the coordinates of point R on the y-axis.
Since PR = QR, the distance between P and R should also be 10.308.
The x-coordinate of point R will be 0 since it lies on the y-axis.
Using the distance formula, we can find the y-coordinate of point R.
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of P (3,7) and point R (0,y), we have:
10.308 = sqrt((0 - 3)^2 + (y - 7)^2)
10.308 = sqrt(9 + (y - 7)^2)
103.07 = 9 + (y - 7)^2
(y - 7)^2 = 94.07
(y - 7) = sqrt(94.07) or (y - 7) = -sqrt(94.07)
y - 7 = 9.699 or y - 7 = -9.699
y = 16.699 or y = -2.699
So, the coordinates of point R on the y-axis such that PR = QR are (0, 16.699) and (0, -2.699).
To find the coordinates of point R, we need to find the midpoint of the line segment PQ. The midpoint is the point that divides the line segment into two equal parts.
The coordinates of the midpoint M(x_m, y_m) can be found using the formula:
x_m = (x_p + x_q) / 2
y_m = (y_p + y_q) / 2
where (x_p, y_p) are the coordinates of point P and (x_q, y_q) are the coordinates of point Q.
Let's substitute the given values into the formula:
x_m = (3 + 11) / 2 = 14 / 2 = 7
y_m = (7 + -6) / 2 = 1 / 2 = 0.5
So, the midpoint of the line segment PQ is M(7, 0.5).
Next, we need to find the equation of the line passing through points P and Q. This line can be represented by the equation of a straight line: y = mx + c, where m is the slope and c is the y-intercept.
Let's calculate the slope m using the formula:
m = (y_q - y_p) / (x_q - x_p)
Substituting the given values:
m = (-6 - 7) / (11 - 3) = -13 / 8
So, the slope of the line passing through points P and Q is -13/8.
Now, we can find the equation of this line using the slope-intercept form. We have the slope m and a point (x_p, y_p) on the line, which is point P(3, 7). The equation is given by:
y - y_p = m(x - x_p)
Substituting the values:
y - 7 = (-13/8)(x - 3)
Simplifying the equation:
y - 7 = (-13/8)x + 39/8
y = (-13/8)x + 39/8 + 7
y = (-13/8)x + 39/8 + 56/8
y = (-13/8)x + 95/8
Now, we need to find the point R on the y-axis such that PR = QR. Since the y-coordinate of point R is 0, we can substitute y = 0 in the equation of the line:
0 = (-13/8)x + 95/8
Solving for x, we get:
(13/8)x = 95/8
x = (95/8) * (8/13)
x = 95/13
So, the x-coordinate of point R is 95/13.
Therefore, the coordinates of point R on the y-axis such that PR = QR are R(95/13, 0).