Multiple choice

If the vectors $AB=3\hat { i } +4\hat { k } $, and $AC=5\hat { i } -2\hat { j } +4\hat { k } $ are the sides of a $\triangle ABC$, then the length of the median through $A$ is

  1. $\sqrt { 18 } $
  2. $\sqrt { 72 } $
  3. $\sqrt { 33 } $
  4. $\sqrt { 45 } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The median through A is the vector from A to the midpoint of BC. Midpoint M of BC is (B+C)/2. Since AB and AC are vectors from A, B = AB and C = AC. M = (AB + AC) / 2 = (3i + 4k + 5i - 2j + 4k) / 2 = (8i - 2j + 8k) / 2 = 4i - j + 4k. Length = sqrt(4^2 + (-1)^2 + 4^2) = sqrt(16 + 1 + 16) = sqrt(33).

AI explanation

In triangle ABC, the median from vertex A goes to the midpoint M of side BC. The vector BC is found by subtracting vector AB from vector AC, resulting in 2i - 2j. The midpoint M is located at half of this vector from B, so vector BM equals i - j. The median vector AM is the sum of vectors AB and BM, yielding 4i - j + 4k. The length of the median is the square root of (4 squared plus (-1) squared plus 4 squared), which is the square root of (16 + 1 + 16) or the square root of 33. The correct answer is the square root of 33.

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