geometry question?

Question

justify if the following statement is always, sometimes or never true
In triangle PQR, PQ
I am thinking always true Theorem5.8 but i might be wrong
PLEASE HELP

Answer 1

In triangle PQR
The equation for 3 sides of triagnles is as below:
PQ^2 = PR^2 + RQ^2 - 2(PR)(RQ)cos A
where A is angle between PR and RQ.

PQ^2 = PR^2 + RQ^2 - 2(PR)(RQ)cos A
cosA = (PR^2 + RQ^2 - PQ^2)/(2(PR)(RQ))

Since cosA > -1 and A < 180 degree for triangle
(PR^2 + RQ^2 - PQ^2)/(2(PR)(RQ)) > -1
PR^2 + RQ^2 - PQ^2 > -2(PR)(RQ)
PQ^2 < PR^2 + RQ^2 + 2(PR)(RQ)
PQ^2 < (PR + RQ) ^2
PQ < PR + RQ

it is true!!!

Just to add on.
Since cosA < 1 and A > 0 degree for triangle
(PR^2 + RQ^2 - PQ^2)/(2(PR)(RQ)) < 1
PR^2 + RQ^2 - PQ^2 < 2(PR)(RQ)
PQ^2 > PR^2 + RQ^2 - 2(PR)(RQ)
PQ^2 > (PR - RQ) ^2
PQ > PR - RQ
PR < PQ + RQ

So, any side of triangle is always less than the sum of the other 2 sides of triangle.

Answer 2

yes that is always true
the theorem states
that the sum of any two sides of a triangle is greater than the remaining
it is always true because if PQ = or > PR + RQ then it would be a triangle anymore, itd just b 3 lines which arent joined together

Answer 3

True.

In any triangle, the sum of the two smaller sides must be bigger than the largest side. (Otherwise, they won't reach across to form the triangle).

Therefore, in your example, if PQ is the largest side, then PR + RQ must be bigger.

If PR or RQ is the largest side, then it by itself is larger than PQ, so obviously the sum of the two would be larger than PQ.