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

Question

Tough and Tricky questions: Geometry. https://gmatclub.com/forum/download/file.php?id=25156 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 . 2014-12-30_1850.png

PareshGmat · Accepted Answer

Answer = A. 120º Refer the modified diagram below 2014-12-30_1850.png \angle ADO = \angle AEO = 90 (As circle is inscribed) \angle EOD = 180-60 = 120

littlewarthog · Answer

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.

bankerboy30 · Answer

How do you know he measure of the two are 90?

littlewarthog · Answer

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°.

beattheheat · Answer

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

countonhearts · Answer

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

Bunuel · Answer

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).

GMATWhizTeam · Answer

Since the circle is inscribed inside the triangle, the sides of the triangle form the tangents to the circle. The angle between the tangent and the radius is always 90 degrees.

Also, the line drawn from the external point (from which the tangents are drawn) to the centre of the circle, bisects the angles made at the external point as well as at the centre. 
This means, if A is connected to O with a line segment, 
Angle DAO = Angle EAO and
Angle DOA = Angle EOA

It is also known that angles ODA and OEA are right angles and hence equal to 90 degrees. 
Therefore, triangles ODA and OEA are 30-60-90 right triangles.

Note that angle CAB = angle DAO + angle EAO = 60 degrees. 
Since Angle DAO = angle EAO, each of them will be equal to 30 degrees.

Therefore, Angle DOA = Angle EOA = 60 degrees
Angle EOD = Angle EOA + Angle DOA = 60 + 60 = 120 degrees.

The correct answer option is A.

bumpbot · Answer

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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 =

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