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# In a triangle with sides of lengths 3, 4, and 5, the smallest angle is

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Math Expert
Joined: 02 Sep 2009
Posts: 59589
In a triangle with sides of lengths 3, 4, and 5, the smallest angle is  [#permalink]

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Updated on: 22 Jul 2019, 06:11
00:00

Difficulty:

35% (medium)

Question Stats:

76% (02:26) correct 24% (02:43) wrong based on 37 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

Attachment:

2019-04-04_1007.png [ 18.93 KiB | Viewed 662 times ]

_________________

Originally posted by Bunuel on 03 Apr 2019, 23:08.
Last edited by SajjadAhmad on 22 Jul 2019, 06:11, edited 1 time in total.
Manager
Joined: 20 Oct 2018
Posts: 140
Re: In a triangle with sides of lengths 3, 4, and 5, the smallest angle is  [#permalink]

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06 Apr 2019, 09:59
2
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
Manager
Joined: 11 Aug 2017
Posts: 59
Re: In a triangle with sides of lengths 3, 4, and 5, the smallest angle is  [#permalink]

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12 May 2019, 21:35
aniket16c wrote:
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

WHy cant OAE be smallest triangle? any quick way of understanding the situation?
Manager
Joined: 20 Oct 2018
Posts: 140
Re: In a triangle with sides of lengths 3, 4, and 5, the smallest angle is  [#permalink]

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13 May 2019, 00:27
1

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!
Re: In a triangle with sides of lengths 3, 4, and 5, the smallest angle is   [#permalink] 13 May 2019, 00:27
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