Suppose that R is at (8, 6). Let P be a point on the line 8y = 15x and Q be a point on the line 10y = 3x, and suppose that R is the midpoint of PQ. Then the length of PQ can be written as a / b, where a and b have no common factors other than 1. What is a + b.
If P is on 8y = 15x
then we could call P (p,15p/8)
similarly if Q is on 10y = 3x
then let's refer to Q as (q,3q/10)
given that R(8,6) is the midpoint of PQ
(p+q)/2 = 8
p + q =16 -----> q = 16-p
and ((15p/8 + 3q/10)/2 = 6
15p/8 + 3q/10 = 12
times 40
75p + 12q = 480 or
25p + 4q = 160
25p + 4(16-p) = 160
21p = 96
p= 32/7
P is (32/7 , 60/7)
q = 16-p = 16-32/7 = 80/7
Q is (80/7 , 24/7)
PQ = √(80/7 - 32/7)^2 + (24/7-60/7)^2)
= .. you do the button pushing,
I got a fraction of the form a/b, then do a+b
let P = (m,15/8 m)
Q = (n,3/10 n)
Then since PR=QR,
(8-m)^2 + (6 - 15/8 m)^2 = (8-n)^2 + (6 - 3/10 n)^2
and, since the slope of PR = slope of QR,
(15/8 m - 6)/(m-8) = (3/10 n - 6)/(n-8)
Solving for m and n, we have
m = 32/7, n = 80/7
Plugging that into the distance formula, we see that PQ = 60/7
To solve this problem, we need to find the coordinates of points P and Q, and then use the distance formula to determine the length of line PQ.
Step 1: Find the coordinates of point P.
We know that R is the midpoint of line PQ, which means that the x-coordinate of R will be the average of the x-coordinates of P and Q, and the y-coordinate of R will be the average of the y-coordinates of P and Q.
Given that R is at (8, 6), we can use this information to find the x-coordinate of P. Since the x-coordinate of R is also the average of the x-coordinates of P and Q, we have:
x-coordinate of P = 2 * x-coordinate of R - x-coordinate of Q
Substituting the given values, we get:
x-coordinate of P = 2 * 8 - x-coordinate of Q
However, we do not know the x-coordinate of Q yet. To find it, we can use the equation of the line 10y = 3x. We can rearrange the equation to isolate y:
y = (3/10)x
Now we can substitute this expression for y into the equation of the other line, 8y = 15x, and solve for x:
8 * (3/10)x = 15x
(24/10)x = 15x
(24/10)x - 15x = 0
(24 - 150/10)x = 0
(-126/10)x = 0
x = 0
So, the x-coordinate of Q is 0. Now we can substitute this value into the equation we obtained earlier to find the x-coordinate of P:
x-coordinate of P = 2 * 8 - 0
x-coordinate of P = 16
Therefore, the coordinates of point P are (16, y).
Step 2: Find the coordinates of point Q.
We already know that the x-coordinate of Q is 0. We can substitute this value into the equation of the line 10y = 3x to find the y-coordinate of Q:
10y = 3 * 0
10y = 0
y = 0
Therefore, the coordinates of point Q are (0, 0).
Step 3: Find the length of line PQ.
Now that we have the coordinates of points P (16, y) and Q (0, 0), we can use the distance formula to find the length of line PQ:
Length of line PQ = √[(x2 - x1)^2 + (y2 - y1)^2]
Substituting the values, we get:
Length of line PQ = √[(0 - 16)^2 + (0 - y)^2]
= √[(-16)^2 + (-y)^2]
= √[256 + y^2]
Therefore, the length of line PQ is √[256 + y^2].
Step 4: Expressing the length of PQ as a fraction.
The length of PQ is √[256 + y^2], and we know that R is the midpoint of PQ, which means that the x-coordinate of R (8) is the average of the x-coordinates of P and Q, and the y-coordinate of R (6) is the average of the y-coordinates of P and Q.
Using the equation for the x-coordinate of P we found earlier, we have:
16 = 2 * 8 - x-coordinate of Q
16 = 16 - x-coordinate of Q
x-coordinate of Q = 0
Using the equation for the y-coordinate of P, we have:
6 = 0 + y-coordinate of Q
y-coordinate of Q = 6
So, the coordinates of point Q are (0, 6).
Substituting the values of x and y into the equation for the length of line PQ, we get:
Length of line PQ = √[256 + (6)^2]
= √[256 + 36]
= √[292]
= √(2 * 2 * 73)
= 2√73
Therefore, the length of line PQ is 2√73, and the sum of the numerator and denominator of the fraction a/b is 2 + 73 = 75.
Therefore, a + b = 75.
To solve this problem, we need to find the coordinates of points P and Q, and then use the distance formula to find the length of PQ.
Step 1: Find the equation of the line passing through R with slope -15/8.
The given line is 8y = 15x. Rewriting it in slope-intercept form, we have:
y = (15/8)x
Since R is on this line, we can substitute R's coordinates (8, 6) into the equation to find the y-intercept:
6 = (15/8)(8)
6 = 15
The equation becomes: y = (15/8)x + 15.
Step 2: Find the equation of the line passing through R with slope -3/10.
The given line is 10y = 3x. Rewriting it in slope-intercept form, we have:
y = (3/10)x
Substituting R's coordinates (8, 6) into the equation to find the y-intercept:
6 = (3/10)(8)
6 = 12/10
The equation becomes: y = (3/10)x + 12/10.
Step 3: Find the coordinates of point P.
Since R is the midpoint of PQ, we know that the x-coordinate of P is the average of the x-coordinates of R and Q.
(8 + x-coordinate of Q)/2 = x-coordinate of P
(8 + x-coordinate of Q)/2 = 8
8 + x-coordinate of Q = 16
x-coordinate of Q = 16 - 8
x-coordinate of Q = 8
Substituting the x-coordinate of Q into either equation from Step 1 or Step 2, we can find the y-coordinate of point P.
Using the equation y = (15/8)x + 15:
y = (15/8)(8) + 15
y = 15 + 15
y = 30
Therefore, the coordinates of point P are (8, 30).
Step 4: Find the coordinates of point Q.
Since R is the midpoint of PQ, we know that the x-coordinate of Q is the average of the x-coordinates of R and P.
(8 + x-coordinate of P)/2 = x-coordinate of Q
(8 + 8)/2 = x-coordinate of Q
16/2 = x-coordinate of Q
x-coordinate of Q = 8
Substituting the x-coordinate of Q into either equation from Step 1 or Step 2, we can find the y-coordinate of point Q.
Using the equation y = (15/8)x + 15:
y = (15/8)(8) + 15
y = 15 + 15
y = 30
Therefore, the coordinates of point Q are (8, 30).
Step 5: Use the distance formula to find the length of PQ.
The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's substitute the coordinates of P and Q into the formula:
d = sqrt((8 - 8)^2 + (30 - 6)^2)
d = sqrt(0 + 24^2)
d = sqrt(576)
d = 24
Therefore, the length of PQ is 24.
Step 6: Find a + b.
The length of PQ can be written as a / b, where a = 24 and b = 1 (since any non-zero number can be divided by 1).
Therefore, a + b = 24 + 1 = 25.
So, a + b = 25.
Well, it seems like we have ourselves a geometry problem! Let's solve it with a touch of humor, shall we?
First, let's find the coordinates of point P. We know that R is at (8, 6), and since R is the midpoint of PQ, we can use the midpoint formula to find P.
So let's set up our formula:
(x₁ + x₂)/2 = x₃ and (y₁ + y₂)/2 = y₃
We know that R is at (8, 6), which means x₁ = 8 and y₁ = 6.
Now, let's find the slope of the line 8y = 15x.
Dividing both sides by 8, we get y = (15/8)x.
So the slope is (15/8). But since the line is perpendicular to 8y = 15x, the slope of the line PQ is the negative reciprocal of (15/8).
The negative reciprocal would be -8/15. So the slope of the line PQ is -8/15.
Now we can use the point-slope formula to find the equation of line PQ. Since we know R (8, 6) is on the line, we can use that as one of the points.
Using the formula y - y₁ = m(x - x₁), we get:
y - 6 = (-8/15)(x - 8)
Expanding that out, we get:
y - 6 = (-8/15)x + (64/15)
Now, we can rewrite the equation in standard form, Ax + By = C:
(8/15)x + y = 6 + (8/15)(8)
Multiplying through by 15, we get:
8x + 15y = 90 + 64
8x + 15y = 154
So the equation of PQ is 8x + 15y = 154.
Now, we need to find where this line intersects with the line 10y = 3x.
Substituting 10y = 3x into the equation of PQ, we get:
8x + 15(3x/10) = 154
Multiplying through by 10, we get:
80x + 45x = 1540
125x = 1540
x = 1540/125
Simplifying, we get:
x = 1232/100
And now we can substitute this value of x into the equation 10y = 3x to find the value of y:
10y = 3(1232/100)
10y = 3696/100
y = 3696/1000
Now, using the distance formula, we can find the distance between points P and Q.
The distance formula can be written as:
√((x₂ - x₁)² + (y₂ - y₁)²)
Using the coordinates of P (x₂, y₂) and R (x₁, y₁), we can find the length of PQ.
The length of PQ is:
√((1232/100 - 8)² + (3696/1000 - 6)²)
Simplifying this expression, we get:
√(((1232 - 800)/100)² + ((3696 - 6000)/1000)²)
√((432/100)² + (-2304/1000)²)
√(18/25 + 1296/100000)
√(720/100000)
√(9/1250)
So the length of PQ is √(9/1250).
That means a = 9 and b = 1250.
And finally, a + b = 9 + 1250 = 1259.
So, the value of a + b is 1259. Ta-da!