Answer:
(a) It is an isosceles triangle with sides of lengths: 6, 2√10, and 6 units.
(b) It is a right-angled triangle with hypotenuse side of 3√5 units, and opposite and adjacent sides of 6 and 3 units
Explanation:
(a) P(3,−2,−3), Q(7,0,1), R(1,2,1)
|PQ| = √[(7-3)² + (0+2)² + (1+3)²]
= √(4² + 2² + 4²)
= √36
= 6
|QR| = √[(1-7)² + (2-0)² + (1-1)²]
= √((-6)² + 2² + 0)
= √40
= 2√10
|RP| = √[(3-1)² (-2-2)² (-3-1)²]
= √[(2² + (-4)² + (-4)²]
= √36
= 6
|PQ| = |RP|
So, triangle is isosceles.
(b) P(2,−1,0), Q(4,1,1), R(4,−5,4)
|PQ| = √[(4-2)² + (1+1)² + (1-0)²]
= √(2² + 2² + 1²)
= √9
= 3
|QR| = √[(4-4)² (-5-1)² (4-1)²]
= √[(0 + (-6)² + 3²]
= √45
= 3√5
|RP| = √[(2-4)² + (-1+5)² + (0-4)²]
= √((-2)² + 4² + (-4)²)
= √36
= 6
|PQ|² + |RP|² = |QR|²
So, triangle is right-angled