3D Geometry
Section Formula
Grade 12
Question:
<p>If \(A(3, 2, 0)\), \(B(5, 3, 2)\) and \(C(-9, 6, -3)\) are three points forming a triangle and \(AD\) is the bisector of \(\angle BAC\), then coordinates of \(D\) are</p>
<p>\(\left(\dfrac{17}{16}, \dfrac{57}{16}, \dfrac{28}{16}\right)\)</p>
<p>\(\left(\dfrac{38}{16}, \dfrac{57}{16}, \dfrac{17}{16}\right)\)</p>
<p>\(\left(\dfrac{38}{16}, \dfrac{17}{16}, \dfrac{57}{16}\right)\)</p>
<p>\(\left(\dfrac{57}{16}, \dfrac{38}{16}, \dfrac{17}{16}\right)\)</p>
Step-by-Step Solution
Key Concept: The angle bisector theorem states that D divides BC in the ratio AB:AC. Calculate these distances using the distance formula, then use the section formula to find D's coordinates.
Step 1: Find distances AB and AC using the distance formula. AB = √[(5-3)^2 + (3-2)^2 + (2-0)^2] = √[4 + 1 + 4] = √9 = 3 AC = √[(-9-3)^2 + (6-2)^2 + (-3-0)^2] = √[144 + 16 + 9] = √169 = 13 Step 2: By the angle bisector theorem, D divides BC in the ratio AB:AC = 3:13. Step 3: Use the section formula. If D divides BC in ratio 3:13, then: D = [(3·C + 13·B)/(3+13)] = [(3(-9, 6, -3) + 13(5, 3, 2))/16] D = [(-27 + 65, 18 + 39, -9 + 26)/16] = [(38, 57, 17)/16] D = (38/16, 57/16, 17/16) = (19/8, 57/16, 17/16) ∴ Answer: D = (19/8, 57/16, 17/16) or equivalently (2.375, 3.5625, 1.0625)
Correct Answer: A