Question 12 of 13intermediate🔧 ApplyNumerical4 marks

If the origin is the centroid of the triangle PQR with vertices P(2a, 2, 6), Q(−4, 3b, −10) and R(8, 14, 2c), then find the values of a, b and c.

Correct Answer

a = -2, b = -16/3, c = 2

Exercise: Miscellaneous Exercise on Chapter 11 | Q: 3 | (Page Page 8)
For More Understanding

Explanation

The centroid formula from Example 9 in the context shows that for a triangle with vertices A, B, C, the centroid G has coordinates ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3). Since the origin (0,0,0) is the centroid, each coordinate sum must equal zero.

Solution Steps

  1. Step 1: Let the centroid G be at origin (0, 0, 0). For triangle PQR with vertices P(2a, 2, 6), Q(−4, 3b, −10) and R(8, 14, 2c), the centroid coordinates are given by:

    x-coordinate: (2a − 4 + 8)/3 y-coordinate: (2 + 3b + 14)/3 z-coordinate: (6 − 10 + 2c)/3

  2. Step 2: Since centroid is at origin (0, 0, 0), equate each coordinate to 0:

    For x-coordinate: (2a − 4 + 8)/3 = 0 ⇒ 2a + 4 = 0 ⇒ a = −2

    For y-coordinate: (2 + 3b + 14)/3 = 0 ⇒ 3b + 16 = 0 ⇒ b = −16/3

    For z-coordinate: (6 − 10 + 2c)/3 = 0 ⇒ 2c − 4 = 0 ⇒ c = 2

  3. Step 3: Therefore, the values are a = −2, b = −16/3, and c = 2.

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