progressions

Question

If a, b, c as well as p, q, r are in geometric progression with the same common ratio, then the points
(a,p), (b,q), (c,r) are:

a. vertices of equilateral triangle
b. vertices of a right angled triangle
c. vertices of a right angled isosceles triangle
d. are collinear

Answer 1

Since a, b, and c are in geometric progression, b = aR, and c = aR², R is the common ratio. 

 

Similarly, q = pR, and r = pR² 

 

So, (b,q) = (aR, pR) = R(a,p)  and (c,r) = (aR², pR²) = R²(a,p)

 

Since (b,q) and (c,r) are both multiples of (a,p), the points (a,p), (b,q), and (c,r) lie on the same line (i.e., the points are collinear). 

 

Answer 2

Hi Akasapu,

 

The answer is D. They should be collinear, that is the points are positioned on the same line. 

Answer 3

I believe I just answered your question from 3 hours ago. Please refer to that, and it sometimes helps to draw the points so you can see how they lie.

It helps if you plug in numbers to visualize the problem. Being in geometric progression, I could choose a common ratio of 3 and: a = 2, b = 6, c = 18, p = 5, q = 15, r = 45 So (2,5) would lie on the same line as (6,15) and (18,45) because the gradient (0.4) is the same. The definition of collinear points is three or more points that lie on a single straight line. So, "d."