SOLUTION: The coordinates of the point P,Q,R are (-1,11),(2,5) and (t,3) respectively. Given that ∠PQR = 90 °, calculate the value of t. The line PQ is produced to S so that QS = PQ. Calc

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Question 1178012: The coordinates of the point P,Q,R are (-1,11),(2,5) and (t,3) respectively. Given that ∠PQR = 90 °, calculate the value of t. The line PQ is produced to S so that QS = PQ. Calculate the coordinates of S.
Found 3 solutions by Solver92311, MathLover1, greenestamps:
Answer by Solver92311(821) About Me  (Show Source):
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If the measure of angle PQR is 90°, then the slope of segment PQ must be the negative reciprocal of segment QR.

Use the slope formula:



to calculate the slope of segment PQ. Calculate the negative reciprocal of the slope of segment PQ. Use the slope formula for segment QR with as a variable in the formula, set the formula equal to the negative reciprocal of the slope of segment PQ, and then solve for .

John

My calculator said it, I believe it, that settles it

From
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Answer by MathLover1(20850) About Me  (Show Source):
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P,Q,R are (-1 ,11 ),(2 ,5 ) and (t ,3 ) respectively

first find the slope of the line passes through
P(-1 ,11 ) and Q(2 ,5 )
m=%285-11%29%2F%282-%28-1%29%29=-6%2F3=-2
y=-2x%2Bb....use one point to calculate b
5=-2%2A2%2Bb
b=5%2B4
b=9
y=-2x%2B9.....eq.1

since given that ∠PQR+=+90 °, find a line that is perpendicular to line with a slope -2 and passes through Q and R
y=mx%2Bb
the slope of the perpendicular line will be negative reciprocal of the slope -2 and it is
m=-1%2F-2=1%2F2
y=%281%2F2%29x%2Bb...use one point to calculate b
5=%281%2F2%292%2Bb
5=1%2Bb
b=4
y=%281%2F2%29x%2B4.....eq.2
since given (t ,3 ), we use line+y=3 and the intersection point of the y=%281%2F2%29x%2B4 and +y=3 will have x=t
y=%281%2F2%29x%2B4...substitute +y
3=%281%2F2%29x%2B4
3-4=%281%2F2%29x
-1%2A2=x
x=-2=>t=-2 and the point R(-2 ,3 )

given also: the line+PQ is produced to S so that QS+=+PQ. Calculate the coordinates of S.
S will be on a line y=%281%2F2%29x%2B4, from Q same distance as +PQ

d=sqrt%28%282-%28-1%29%29%5E2%2B%285-11%29%5E2%29
d=sqrt%283%5E2%2B%28-6%29%5E2%29
distance is d=sqrt%2845%29=6.7 which is radius r of the circle that passes through P, Q, and S
use the radius , draw a circle and read from the graph what are the coordinates of the point S




S(-4 ,2 )

Answer by greenestamps(13200) About Me  (Show Source):
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Part 1: find t.

From P to Q is 3 units right and 6 units down; the slope of PQ is -2.

PQR is a right angle, so the slope of QR is +1/2. That means on QR the change in y is 1/2 the change in x -- i.e., the change in x is 2 times the change in y.

From Q to R the change in y is -2, so the change in x has to be -4. A change of -4 in x and a change of -2 in y from Q(2,5) puts us at R(-2,3).

ANSWER: t = -2.

Part 2: Find the coordinates of point S.

S is on PQ extended, and QS = PQ.

From P to Q was 3 units right and 6 units down; from Q to S will be the same.

3 right and 6 down from Q(2,5) puts us at S(5,-1).

ANSWER: S is (5,-1).