Question

There are three points P, Q and R on a straight line such that PQ : QR = 3:5. If n is the number of possible values of PQ : PR, then what is n equal to ?​

Answer 1

Answer:

thought I'd create some images to go along with Carcass' solution.

Let's first consider 2 points (P and Q)



One obvious point that is equidistant from P and Q is this point...



Also recognize that, if we draw two circles of equal radii with their centers at PQ, the 2 points of intersection will also be equidistant from P and Q



In fact, all points on this red line are equidistant from P and Q. The red line (as Carcass mentioned above) is the perpendicular bisector of points P and Q.



At this point, let's add a 3rd point (R) and draw the perpendicular bisector of points R and P in blue. All points on the blue line will be equidistant from R and P.



So, the green point (where the blue line and red line intersect) must be equidistant from R and P, AND it must equidistant from P and Q.

So, that green point must be equidistant from P, Q and R

Since there is only 1 such green point, the correct answer is B.

Cheers,

Brent

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Most Helpful Community Reply

bessonov

Aug 9, 2019

Another useful way to think about this problem is to realize that three points determine a circle. We can draw a circle such that points P, Q, and R all lie on the circumference, with the circle's center being the same distance (radius) from all three points. Since the points are not colinear only one such circle is possible.

Answer 2

Given : There are three points P, Q and R on a straight line such that PQ : QR = 3:5.

n is the number of possible values of PQ : PR,

To Find :  Value of n

Solution:

PQ : QR = 3:5.

PQ = 3x  and QR = 5x

Case 1  : Q is in between P and R

P    ---3x --- Q  --- 5x  ---- R

PR = PQ + QR = 3x + 5x = 8x

PQ : PR = 3x : 8x  

=> PQ:QR  = 3 : 8

Case 2 :

R is between P and  Q

not possible  as QR > PQ

Case 3 :

P is between Q and R

Q --- 3x  --- P  --2x  -- R

QR = PQ + PR

=> 5x = 3x + PR

=> PR = 2x

PQ : QR  =  3x : 2x   = 3: 2

PQ:QR  = 3 : 8  or  3:2

Hence two possible values

n = 2

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