Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isoseceles triangle?(a)P(3,−2,−3),Q(7,0,1),R(1 ,2,1)(b)P(2,−1,0),Q(4,1,1),R(4,−5,4 )

(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

Step-by-step 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

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