In the following examples, can the segment joining the given points form a triangle ? If triangle is formed, state the type of the triangle considering sides of the triangle.
(2) P(-2,-6) , Q(-4,-2), R(-5,0)
8. In the following examples, can the segment joining the given points form a triangle ? If triangle is formed, state the type of the triangle considering sides of the triangle.
(2) P(-2,-6) , Q(-4,-2), R(-5,0)
Solution
P(–2,–6) , Q(–4,–2), R(–5,0)
[PQ = sqrt{left( – 2 + 4 right)^2 + left( – 6 + 2 right)^2} = sqrt{4 + 16} = sqrt{20} = 2sqrt{5}]
[QR = sqrt{left( – 4 + 5 right)^2 + left( – 2 + 0 right)^2} = sqrt{1 + 4} = sqrt{5}]
[PR = sqrt{left( – 2 + 5 right)^2 + left( – 6 + 0 right)^2} = sqrt{9 + 36} = sqrt{45} = 3sqrt{5}]
[PQ + QR = PR]
Since the sum of the two sides is not greater than the third side so, the given vertices do not form a triangle.
Explanation:-
The given points are P(-2,-6), Q(-4,-2), and R(-5,0).
Using the distance formula, we can find the lengths of the sides of the triangle:
- PQ = √[(−4+2)²+(−2+6)²] = √20 = 2√5
- QR = √[(−5+4)²+(0+2)²] = √5
- PR = √[(−5+2)²+(0+6)²] = √45 = 3√5
The sum of the lengths of PQ and QR is 2√5 + √5 = 3√5, which is equal to the length of PR.
However, this violates the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, PQ + QR = 3√5 is not greater than PR = 3√5, so the given vertices do not form a triangle.
Chapter 5. Co-ordinate Geometry – Problem set 5 (Page 122)