Rainman91 wrote:
A question. If the cone is right circular one, shouldn't the triangle shown above be equilateral with 60 degrees each? That is why I thought it would be a 90-60-30 triangle. Someone please help.
Hi
Rainman91 Thanks for your query.
Firstly, let me tell you the definition of a right circular cone.
- - A right circular cone is a cone in which the vertex of the cone touches its circular base perpendicularly at the center of the base.
This is all that the definition of a right circular cone tells us. Now, once you used the extra info that this cone is inscribed in a hemisphere, how did you infer that the triangle formed is equilateral? In fact, as I will prove to you soon, this big triangle is 45-45-90.
PROOF:As per the question, “
A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere”.
The image below does 100% justice to every piece of information given in the question. We can see that the
height of the cone equals the
radius of the cone/hemisphere.
DEEP DIVE INTO THE TRIANGLES FORMED:Now, let us focus only on triangles formed. Look at the figure below, where I removed everything other than the triangles.
So, firstly, let’s observe the triangle ABC,
- We know that ∠ B measures 90 degrees (from the definition of a right circular cone).
- Also, AB = BC = r (radius)
- Now, we know that in any triangle, angles opposite to equal sides are equal.
- Thus, since AB = BC, we get that ∠ A = ∠ C = x (say).
Finally, using the angle sum property in triangle ABC, we get:
- ∠ A + ∠ B + ∠ C = 180°
- x + 90 + x = 180°
- 2x = 90°
- x = 90/2 = 45°.
- Thus, ∠ A = ∠ C = 45° …(I)
Following the same process in triangle ABD, we can say that ∠ D = ∠ A = 45° … (II)
Now, if we combine (I) and (II), we can say in the
big triangle (triangle ACD), we have:
- ∠ C = 45°
- ∠ D = 45°
- And ∠ A (angle CAD) = ∠ CAB + ∠ DAB
That implies, triangle ACD is also a 45 – 45 – 90 triangle, and NOT an equilateral triangle as you assumed.
Hope this helps!
Best,
Aditi Gupta
Quant expert,
e-GMAT