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**jacks****Member**- Registered: 2012-11-21
- Posts: 80

if the vertices P, Q, R of a triangle PQR are rational points

Then which of the following points is(are) always rational point(s)

options.

(a)centroid (b)incentre (c)orthocentre (d)circumcentreplz explain answer

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

hi jacks

Hhhmmm. This is a new problem for me. But I suppose you could tackle it like this:

centroid: is at the intersection of the medians (which join the midpoint of a side to the opposite vertex, and is also one third of the way up any median.

So if P and Q have rational coordinates so will the midpoint of PQ.

R is also rational so the point one third of the way up from PQ towards R will be rational.

How does that sound to you?

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**jacks****Member**- Registered: 2012-11-21
- Posts: 80

Thanks bob bundy

but I did not understand the meaning of the Given line

R is also rational so the point one third of the way up from PQ towards R will be rational

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

Ok. Let's say P is (a,b) Q is (c,d) and R is (e,f)

a,b,c,d,e and f are all rational ie they are fractions

Midpoint PQ

and centroid is

the {rationals} are closed for +, - , x and ÷

ie. adding two fractions, or subtracting one fraction from another, or multiplying two fractions or dividing them will always give another fracftion.

Therefore both the midpoint and the centroid have rational coordinates.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**jacks****Member**- Registered: 2012-11-21
- Posts: 80

Thanks Bobbundy Got it.

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