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As shown in the figure above, line segments AB and AC are ta

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honchos
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As shown in the figure above, line segments AB and AC are ta [#permalink] New post   17 Jul 2014, 19:40
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As shown in the figure above, line segments AB and AC are tangent to circle O. If line segments BD and DA have the same length, what is angle BAO? (Note: Figure not drawn to scale.)

A. 15º
B. 30º
C. 36º
D. 45º
E. 50º
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a13ssandra
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Re: As shown in the figure above, line segments AB and AC are ta [#permalink] New post   17 Jul 2014, 20:46
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I did't think this question was easy, but I finally came up with the solution.

step 1: since AB and AC are tangent to the circle then, angles OBA = OCA =90º
step 2: Angle BAO = angle DBA = x and angle BDA = y
step 3: OB = OD (both radii of circle O) and therefore triangle BOD is isosceles so angle OBD = angle ODB =z
step 4: 2x + y = 180 , y + z =180 , x+z =90
step 5: solve both equations for variable z -> z=180-y, z = 90-x
step 6: 90-x=180-y
y-x = 90
step 7: y+2x=180
y - x = 90 (-1)
3x=90
x = 30

If someone knows a faster way to solve this question please post!
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As shown in the figure above, line segments AB and AC are ta [#permalink] New post   Updated on: 20 Jul 2014, 19:29
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It seems to me that if a line drawn from the hypotenuse of a right triangle to the opposite vertex, creates two triangles, such that one of the them is isosceles, the other has to be an isosceles or an equilateral triangle. We can easily arrive at the answer if that is the case.

Originally posted by SVaidyaraman on 17 Jul 2014, 22:37.
Last edited by SVaidyaraman on 20 Jul 2014, 19:29, edited 1 time in total.
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Re: As shown in the figure above, line segments AB and AC are ta [#permalink] New post   20 Jul 2014, 17:49
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a13ssandra wrote:
I did't think this question was easy, but I finally came up with the solution.

step 1: since AB and AC are tangent to the circle then, angles OBA = OCA =90º
step 2: Angle BAO = angle DBA = x and angle BDA = y
step 3: OB = OD (both radii of circle O) and therefore triangle BOD is isosceles so angle OBD = angle ODB =z
step 4: 2x + y = 180 , y + z =180 , x+z =90
step 5: solve both equations for variable z -> z=180-y, z = 90-x
step 6: 90-x=180-y
y-x = 90
step 7: y+2x=180
y - x = 90 (-1)
3x=90
x = 30

If someone knows a faster way to solve this question please post!


No need to use another variable z in step 3, apply the theorem "exterior angle = sum of opp. interior angle".
angle ODB (=OBD) = angle BAO + angle DBA = 2x
angle OBD + angle DBA = 90
\(2x + x = 90\)
\(x = 30\)
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Re: As shown in the figure above, line segments AB and AC are ta [#permalink] New post   20 Jul 2014, 21:30
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SravnaTestPrep wrote:
It seems to me that if a line drawn from the hypotenuse of a right triangle to the opposite vertex, creates two triangles, such that one of the them is isosceles, the other has to be an isosceles or an equilateral triangle. We can easily arrive at the answer if that is the case.


In the above case 2OBD = BDA
We also have OBD+DBO= BDA
therefore OBD=DBO and so triangle OBD is equilateral from which we can find BAD=BAO=30 degrees
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As shown in the figure above, line segments AB and AC are ta [#permalink] New post   25 Aug 2015, 03:59
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honchos wrote:
Attachment:
geometry_graphics_1.gif
As shown in the figure above, line segments AB and AC are tangent to circle O. If line segments BD and DA have the same length, what is angle BAO? (Note: Figure not drawn to scale.)

A. 15º
B. 30º
C. 36º
D. 45º
E. 50º


Angle OBA = 90

Angle ODB = Angle DBA + Angle DAB ---------- (Exterior angle = Sum of opposite interior angles)

We can say ODB is double the angle DBA as the triangle ADB is isosceles ................. Which In turn means Angle OBD is double the angle DBA As the triangle BOD is isosceles

Considering above Angle OBA = 90 can only be split in 30 + 60 ....... so the angle OAB = 30
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Re: As shown in the figure above, line segments AB and AC are ta [#permalink] New post   22 Nov 2019, 23:57
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Can you please explain
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Re: As shown in the figure above, line segments AB and AC are ta [#permalink] New post   30 Sep 2021, 11:49
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Re: As shown in the figure above, line segments AB and AC are ta [#permalink]
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