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Question 1139586: find y if the distance between points p and R is 25 and point R is located in the first quadrant.
P=(3,-8)
R=(10,y)
Found 4 solutions by ikleyn, MathLover1, Theo, greenestamps:
Answer by ikleyn(52776)   (Show Source):
Answer by MathLover1(20849)  (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! point p is at (3,-8)
point r is at (10,y)
the distance between them is 25.
point r is in the first quadrant.
this means that y has to be positive.
the distance between point p and point r is equal to sqrt((y+8)^2 + (10-3)^2).
simplify this to get distance between points p and r is equal to sqrt((y+8)^2 + 49)
since the distance between points p and r is 25, then the formula becomes:
25 = sqrt((y+8)^2 + 49)
square both sides of the equation to get 625 = (y+8)^2 + 49
simplify to get 625 = y^2 + 16y + 64 + 49
combine like terms to get 625 = y^2 + 16y + 113
subtract 625 from both sides of the equation to get 0 = y^2 + 16y - 512.
factor this quadratic equation to get (y + 32) * (y - 16) = 0
solve for y to get y = -32 or 16.
y is positive, so y has to be 16.
your solution is that y = 16.
this means that point p = (3,-8) and point r = (10,16)
the distance between points p and r is equal to sqrt((16+8)^2 + (10-3)^2).
that becomes equal to sqrt((24)^2 + 7^2) which becomes equal to sqrt(625) which becomes equal to 25.
that confirms that, when y = 16, the distance between p and r is 25.
the equation of the line between points p and r is y = 24/7 * x -128/7.
the graph of that equation is shown below.
it shows that the points (3,-8) and (10,16) are both on the line, as they sh ould be.
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
The distance between the two points is 25; the difference between the x-coordinates is 7.
7-24-25 is a Pythagorean triple, so the difference between the y-coordinates must be 24.
If R is to be in the first quadrant, y must be -8+24 = 16.
ANSWER: y = 16
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