TS EAMCET · Maths · Three Dimensional Geometry

If $A (3, 2, 3), B (1, 4, 6)$ and $C (7, 4, 5)$ are the three vertices of a parallelogram $A B C D$ , then the angle between its diagonal through $D$ and the side $D C$ is

A $\cos^{- 1} (\frac{16}{\sqrt{357}})$
B $\cos^{- 1} (\frac{5}{\sqrt{126}})$
C $\cos^{- 1} (\frac{5}{\sqrt{21}})$
D $\cos^{1} (\frac{2}{\sqrt{357}})$

Verified Solution

Answer & Solution

Correct Answer

(A) $\cos^{- 1} (\frac{16}{\sqrt{357}})$

Step-by-step Solution

Detailed explanation

M is the midpoint of AC and BD M3+72,2+42,3+52=M5,3,4 Now using diagonal BD x+12=5 ⇒x=9 y+42=3⇒y=2 z+62=4⇒z=2 Direction ratios of DC, a1=2, b1=-2, c1=-3 Direction ratios of DB, a2=8, b2=-2, c2=-4 Angle…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If A(3,2,3), B(1,4,6) and C(7,4,5) are the three vertices of a parallelogram ABCD, then the angle between its diagonal through D and the side DC is

Step-by-step Solution

If $A (3, 2, 3), B (1, 4, 6)$ and $C (7, 4, 5)$ are the three vertices of a parallelogram $A B C D$ , then the angle between its diagonal through $D$ and the side $D C$ is