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

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

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Tough and Tricky questions: Geometry.



Image
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.

[Reveal] Spoiler:
Attachment:
2014-12-30_1850.png
2014-12-30_1850.png [ 4.77 KiB | Viewed 2508 times ]
[Reveal] Spoiler: OA

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

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New post 30 Dec 2014, 07:34
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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New post 30 Dec 2014, 07:58
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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New post 30 Dec 2014, 08:05
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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Answer = A. 120º

Refer the modified diagram below

Attachment:
2014-12-30_1850.png
2014-12-30_1850.png [ 5.27 KiB | Viewed 2147 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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New post 30 Dec 2014, 21:52
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
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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Tough and Tricky questions: Geometry.



Image
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.

[Reveal] Spoiler:
Attachment:
2014-12-30_1850.png



Answer A. 120.

In Quadrilateral ADOC
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

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

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New post 08 Jan 2015, 08:27
Bunuel wrote:

Tough and Tricky questions: Geometry.



Image
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.

[Reveal] Spoiler:
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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New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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

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Re: A circle is inscribed inside a right triangle as shown. If angle CAB =   [#permalink] 26 Dec 2017, 02:33
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