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555-605 (Medium)|   Geometry|      
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Bunuel
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In a triangle with sides of lengths 3, 4, and 5, the smallest angle is Updated on: 22 Jul 2019, 06:11
Originally posted by Bunuel on 03 Apr 2019, 23:08.
Last edited by Sajjad1994 on 22 Jul 2019, 06:11, edited 1 time in total.
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

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35% (medium)

Question Stats:

74% (02:19) correct 26% (02:18) wrong based on 65 sessions

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In a triangle with sides of lengths 3, 4, and 5, the smallest angle is 36.87°. In the figure, O is the center of the circle of radius 5. A and B are two points on the circle, and the distance between the points is 6. What is the value of x ?

(A) 36.87

(B) 45

(C) 53.13

(D) 116.86

(E) 126.86

Source: Nova GMAT
Difficulty Level: 600
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C
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In a triangle with sides of lengths 3, 4, and 5, the smallest angle is 06 Apr 2019, 09:59
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Theorem: Perpendicular drawn from the center of the circle to a chord bisects the chord.

Drop a perpendicular from center to chord AB. It will bisect the chord. Consider the perpendicular intersects the chord at point E.
Triangle AOE = Right angled triangle with right angle at E.
Angle opposite to smallest side = smallest angle. Angle AOE = 36.87
Angle OAE = x = 90-36.87=53.13
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Mohammad Ali Khan
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In a triangle with sides of lengths 3, 4, and 5, the smallest angle is 12 May 2019, 21:35
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aniket16c
Theorem: Perpendicular drawn from the center of the circle to a chord bisects the chord.

Drop a perpendicular from center to chord AB. It will bisect the chord. Consider the perpendicular intersects the chord at point E.
Triangle AOE = Right angled triangle with right angle at E.
Angle opposite to smallest side = smallest angle. Angle AOE = 36.87
Angle OAE = x = 90-36.87=53.13
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WHy cant OAE be smallest triangle? any quick way of understanding the situation?
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In a triangle with sides of lengths 3, 4, and 5, the smallest angle is 13 May 2019, 00:27
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mohammad Ali Khan:

Let me explain the solution in more detail. This may help you resolve your doubt.
Radius of circle = 5. Thus OA=OB=5.
When we drop a perpendicular from O to AB --> It will bisect AB --> AE = BE= 3.
Thus we get a 3-4-5 right angle triangle. --> AE=3, EO=4, OA=5
Using the theorem: Angle opposite to smallest side = smallest angle --> we can conclude that opposite to smallest side (AE) we will have the smallest angle. Thus Angle AOE = 36.87

Hope the explanation clears your doubt!
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In a triangle with sides of lengths 3, 4, and 5, the smallest angle is 19 Mar 2021, 07:23
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Since options are wide apart, we can use approximation too. imagine a triangle with 60, 60, 60 degree angles with side 5,5,5. If we increase a single side ( moving from 5 towards 6 ) then the angle opposite to the increasing side will increase in proportion to the side and two equal angles ( now triangle is iso ) of the triangle will decrease. so angles will be 60+ , 60-, 60 -. We can guess the answer in this case which is closer to 60 and between 50-60. so.. answer would be C.
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