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# A circle is inscribed inside a right triangle as shown. If angle CAB =

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Math Expert
Joined: 02 Sep 2009
Posts: 47978
A circle is inscribed inside a right triangle as shown. If angle CAB =  [#permalink]

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30 Dec 2014, 07:52
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Difficulty:

5% (low)

Question Stats:

83% (00:59) correct 17% (01:36) wrong based on 149 sessions

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Tough and Tricky questions: Geometry.

A circle is inscribed inside a right triangle as shown. If angle CAB = 60º, what is angle EOD? (Note: Figure not drawn to scale.)

A. 120º
B. 135º
C. 140º
D. 150º
E. 155º

Kudos for a correct solution.

Attachment:

2014-12-30_1850.png [ 4.77 KiB | Viewed 3043 times ]

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Re: A circle is inscribed inside a right triangle as shown. If angle CAB =  [#permalink]

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30 Dec 2014, 08:34
The following solution is based on the assumption, that O is the center of the circle. I am not sure if that is save to assume, but I am not sure how to solve the question without that.

Since the points E and D are on the circle, the lines OE and OD are radii of the circle and the angles OEA and ODA must be 90°. Since OEAD is a quadrilateral, the sum of all four angles must be 360°, so the remaining angle EOD must be 120°. Answer A.
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Re: A circle is inscribed inside a right triangle as shown. If angle CAB =  [#permalink]

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30 Dec 2014, 08:58
How do you know he measure of the two are 90?
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Re: A circle is inscribed inside a right triangle as shown. If angle CAB =  [#permalink]

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30 Dec 2014, 09:05
Since the circle is inscribed into the triangle, the lines AC and AB are tangent to the circle. A line that is tangent to a circle must have the same angle to the radius as the circle itself. And that angle is 90°.
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Re: A circle is inscribed inside a right triangle as shown. If angle CAB =  [#permalink]

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30 Dec 2014, 20:35
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Refer the modified diagram below

Attachment:

2014-12-30_1850.png [ 5.27 KiB | Viewed 2586 times ]

$$\angle ADO = \angle AEO = 90$$ (As circle is inscribed)

$$\angle EOD = 180-60 = 120$$
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Re: A circle is inscribed inside a right triangle as shown. If angle CAB =  [#permalink]

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30 Dec 2014, 22:52
Radii OD and OE must be perpendicular to the tangents AC and AB respectively.
Thus Angle ODA and Angle OEA = 90 each
Angle DAE + Angle EOD = 180
Angle EOD = 120
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Re: A circle is inscribed inside a right triangle as shown. If angle CAB =  [#permalink]

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02 Jan 2015, 00:20
1
Bunuel wrote:

Tough and Tricky questions: Geometry.

A circle is inscribed inside a right triangle as shown. If angle CAB = 60º, what is angle EOD? (Note: Figure not drawn to scale.)

A. 120º
B. 135º
C. 140º
D. 150º
E. 155º

Kudos for a correct solution.

Attachment:
2014-12-30_1850.png

One angle given angle A = 60 degree.
two tangents 90+90=180
Hence sum of angles in quad =360 deg
180+60 + angle 0 = 360
angle O = 120
Math Expert
Joined: 02 Sep 2009
Posts: 47978
Re: A circle is inscribed inside a right triangle as shown. If angle CAB =  [#permalink]

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08 Jan 2015, 09:27
Bunuel wrote:

Tough and Tricky questions: Geometry.

A circle is inscribed inside a right triangle as shown. If angle CAB = 60º, what is angle EOD? (Note: Figure not drawn to scale.)

A. 120º
B. 135º
C. 140º
D. 150º
E. 155º

Kudos for a correct solution.

Attachment:
2014-12-30_1850.png

OFFICIAL SOLUTION:

(A) Figure AEOD is a quadrilateral, so the sum of its interior angles must be 360º.

Therefore, angle EOD = 360º – angle DAE – angle ADO – angle AEO.

We know that both angles ADO and AEO are equal to 90º since the angle created by a radius and a tangent line is always 90º.

Now we can solve for angle EOD:
Angle EOD = 360º – 60º – 90º – 90º = 120º.

The correct answer is choice (A).
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Re: A circle is inscribed inside a right triangle as shown. If angle CAB =  [#permalink]

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26 Dec 2017, 03:33
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