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Math Expert V
Joined: 02 Sep 2009
Posts: 59587
In the figure shown, O is the centre of the circle. What is the value  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 65% (01:54) correct 35% (02:25) wrong based on 52 sessions

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Competition Mode Question In the figure shown, O is the centre of the circle. What is the value of x+y+z?

A. 104
B. 112
C. 180
D. 188
E. 196

Attachment: Geometry.JPG [ 9.92 KiB | Viewed 703 times ]

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Re: In the figure shown, O is the centre of the circle. What is the value  [#permalink]

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Let the radius of the circle = r
Note that, OA = OB = OC = r
--> triangle OAB & OBC are isosceles
--> ∠A = ∠B = x

In triangle OAB, ∠A + ∠B + 100 = 180
--> 2x + 100 = 180
--> 2x = 80
--> x = 40

Consider the straight line AT,
--> ∠B + y + 108 = 180 (straight angle)
--> x + y = 72
--> y = 72 - x = 72 - 40 = 32

In triangle OBC, ∠B = ∠C = y
--> ∠z + ∠y + ∠y = 180
--> ∠z = 180 - 64 = 116

So, x + y + z = 40 + 32 + 116 = 188

IMO Option D
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Re: In the figure shown, O is the centre of the circle. What is the value  [#permalink]

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In the figure shown, O is the centre of the circle. What is the value of x+y+z?

A. 104
B. 112
C. 180
D. 188
E. 196

From figure we have following:
AO = BO = radius of circle
Thus, x = angle(ABO)
Now x + x + 100 = 180 for triangle(ABO)
x = 40

Similarly,
BO = CO = radius of circle
y = angle(BCO)
Now, y + y + z = 180

Also at point B x + y + 108 = 180 (straight line)
y = 32

Hence 32 + y + z = 180
OR
x + y + z + 32 = 180 + x
x + y + z = 180 + 40 - 32 = 188

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Re: In the figure shown, O is the centre of the circle. What is the value  [#permalink]

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Re: In the figure shown, O is the centre of the circle. What is the value  [#permalink]

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Quote:
In the figure shown, O is the centre of the circle. What is the value of x+y+z?

A. 104
B. 112
C. 180
D. 188
E. 196

2x=180-100
x=40
y=180-108-x
y=72-40=32
z=180-2y
z=180-64=116
x+y+z=40+32+116=188

Ans (D)
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Re: In the figure shown, O is the centre of the circle. What is the value  [#permalink]

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1
A = B= x (as OA and OB are radius)
2x + 100 = 180
X = 40,
X+Y+108 = 180,
Y = 180-148 = 32
2Y + Z = 180, As OB = OC,
Z = 180-64 =116
X+Y+Z = 116+32+40 =188.

Imo. D
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Joined: 10 Oct 2019
Posts: 11
GMAT 1: 400 Q35 V12 Re: In the figure shown, O is the centre of the circle. What is the value  [#permalink]

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1
tingle abo is isosceles as ob and oa are radius of circle , so angle x=40
AT ia 180 so ,Y= 180-40-108=32 , again triangle OBC IS isosceles triangle Z= 180-32-32=116
X+Y+Z=40+32+116=188
ANS:D

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Re: In the figure shown, O is the centre of the circle. What is the value  [#permalink]

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1
From the figure, AO=OB=radius of the circle. This implies, triangle AOB is isosceles, therefore angle OAB (x)= angle ABO.
Hence 2x+100=180 and x=40

Also from the figure, x+y+108=180
which means 40+108+y=180, hence y=180-148=32

OC=OB=radius of the circle. This also implies that triangle BOC is isosceles, therefore, angle OCB=angle OBC (y)
Hence 2y+z=180, hence z+64=180
thus z=180-64=116
x+y+z=40+32+116 = 188

The answer is therefore option D. Re: In the figure shown, O is the centre of the circle. What is the value   [#permalink] 14 Nov 2019, 17:05
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