The diagonal triangle

Geometry Level pending

Let O A OA , O B OB and O C OC be three adjacent edges of a cuboid. Then, which of the following statements could be true.

[1] All the angles in △ A B C \triangle ABC are acute.

[2] △ A B C \triangle ABC is right-angled.

[3] △ A B C \triangle ABC has one obtuse angle in it.

[1] only [1] and [2], but not [3] [1],[2] and [3] [2] and [3], but not [1]

Janardhanan Sivaramakrishnan by Janardhanan Sivaramakris… , 42

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1 solution

Using the fact that O A , O B OA,OB and O C OC are mutually perpendicular, from the figure it can be said that

A B 2 = O A 2 + O B 2 AB^2=OA^2+OB^2 , A C 2 = O C 2 + O A 2 AC^2=OC^2+OA^2 and B C 2 = O B 2 + O C 2 BC^2=OB^2+OC^2 .

For triangle △ A B C \triangle ABC to be a right angled or obtuse angled triangle, if A C AC is the longest side

A B 2 + B C 2 ≤ A C 2 ≤ ( A B + B C ) 2 AB^2+BC^2 \leq AC^2 \leq (AB+BC)^2

Thus, if one of the angles is obtuse in △ A B C \triangle ABC ,

O A 2 + O B 2 + O B 2 + O C 2 = A B 2 + B C 2 ≤ A C 2 = O A 2 + O C 2 OA^2+OB^2+OB^2+OC^2=AB^2+BC^2 \leq AC^2=OA^2+OC^2

This reduces to 2 ( O B ) 2 ≤ 0 2(OB)^2 \leq 0 which is not possible as any length is positive.

Hence, △ A B C \triangle ABC can only be acute angled triangle.

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