How to do this? :(

Answer this questions pls.

1) A line makes angles A,B,C and D with four diagonals of a cube (cosA)^2 + (cosB)^2 + (cosC)^2 + (cosD)^2 = ?

2)A line makes angles A,B,C and D with two diagonals of a cube (cosA)^2 + (cosB)^2 + (cosC)^2 + (cosD)^2 = ?

#Geometry

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Comments

Can you check my solution also?

In #2, can you check if my solution is correct?

The Diagonals of the cube is equal to √(a^2 + b^2 + c^2) = √(3s^2) = s√3

So,

if the diagonals bisect each other, they will just bisect each other.

By using law of cosines in 4 triangles, and also vertical angles are equal, lets say,

<A = <B by vertical angles and also, <C = <D

By Law of cosines,

cos(A) = (b^2 + c - a^2)/(2bc) ----> cos (A) = cos (B) = -1/3

cos (C) = (a^2 + b^2 - c^2)/(2bc) ----> cos (C) = (cos D) = 1/3

therefore,

(cosA)^2 + (cosB)^2 + (cosC)^2 + (cosD)^2= (-1/3)^2 + (-1/3)^2 + (1/3)^2 + (1/3)^2 = 4(1/9) = 4/9

Final Answer: 4/9

Christian Daang - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$