In the above figure, point O is at the center

Question

https://i1168.photobucket.com/albums/r500/GMATPrepNow/zzzz1_zpszf9pyhf3.png

In the above figure, point O is at the center of the circle, and points A, B and C are on the circumference of the circle. 
If ∠BCE = 3xo, what is the measure of ∠ABO?

A) (x + 10)o
B) (180 - x)o
C) (90 - x)o
D) (2x - 30)o
E) (3x - 90)o

*kudos for all correct solutions

chetan2u · Accepted Answer

Hi,

Few geometric rules are useful here..
1) angle ACB= 180- angle BCE=180-3x..
2)   important rule:- angle at centre is TWICE the angle on the circumference from CHORD of SAME length ..
So angle AOB= 2* angle ACB= 2(180-3x)=360-6x..
3) Now Triangle AOB is isosceles triangle, so angle ABO = angle BAO=(180-angleAOB)/2= \frac{180-(360-6x)}{2} =3x-90..
E

rohit8865 · Answer

Please follow below fig.

∠BCA = 180-3x (angle of line)
so ∠AOB = 2(180-3x) (angle subtended on same chord AB)--------(1)
in triangle AOB
sum of all angles are 180
∠ABO + ∠OAB+∠AOB =180
But ∠ABO = ∠OAB =M(let) .... (same radius)---------(2)
substituting values from (1) & (2)
2M + 2(180-3x) =180
M=(3x - 90)º

Ans E

BrentGMATPrepNow · Answer

Given: 
 https://i1168.photobucket.com/albums/r500/GMATPrepNow/zzzz2_zpsn1zlhlhu.png

Since angles on a line must add to 180 degrees, we know that ∠ACB = (180 - 3x) degrees
 https://i1168.photobucket.com/albums/r500/GMATPrepNow/zzzz3_zpswvtq2oe5.png

Since the  inscribed  angle ∠ACB and the  central  angle ∠AOB both contain (hold) the same chord (AB), we know that the central angle is TWICE the inscribed angle. 
In other words, ∠AOB = 2(180 - 3x) = 360 - 6x
 https://i1168.photobucket.com/albums/r500/GMATPrepNow/zzzz4_zps49skl4qu.png

Finally, since OA and OB are radii, we know that these lengths are EQUAL
If these lengths are EQUAL, then ∆AOB is an ISOSCELES triangle, and ∠OAB = ∠ABO
 https://i1168.photobucket.com/albums/r500/GMATPrepNow/zzzz5_zpscl8fudis.png

Let q = ∠OAB = ∠ABO
Since all 3 angles in ∆AOB add to 180 degrees, we can write: (360 - 6x) + q + q = 180
Rearrange: 2q = 6x - 180
Divide both sides by 2 to get: q = 3x - 90
In other words, ∠ABO = 3x - 90

Answer: E

RELATED VIDEO FROM OUR COURSE
 https://www.youtube.com/watch?v=xiyv2MacIUk

sasyaharry · Answer

AB is a chord shared by two triangles: AOB and ACB.
Hence the  \angle AOB = 2* \angle ACB ..................(1)

From the given information  \angle ACB = (180-3x)^ {\circ} ...........(2)

Using (1) and (2): 
 \angle AOB = 2* (180-3x)^ {\circ} ...........(3)

Triangle AOB is an isosceles triangle with OA = OB hence  \angle OAB = \angle OBA 
Since the three angles make up 180, we can write the equation
 \angle OBA + \angle OBA = 180 - 2* (180-3x) 
 2 \angle OBA = 180 - 360 + 6x 
 2 \angle OBA = -180 + 6x 
 2 \angle OBA = 6x - 180 
 2 \angle OBA = \frac{6x - 180}{2} 
 \angle OBA = 3x - 90

Answer is  E

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

In the above figure, point O is at the center

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

In the above figure, point O is at the center

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards