In the diagram above, if arc ABC is a semicircle, what is

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In the diagram above, if arc ABC is a semicircle, what is [#permalink]

26 Aug 2011, 01:24

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In the diagram above, if arc ABC is a semicircle, what is the length of AC?

(1) AD = 2.5
(2) DC = 10

[Reveal] Spoiler: OA

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Last edited by Berbatov on 26 Aug 2011, 01:59, edited 1 time in total.

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Re: Length of AC [#permalink]

26 Aug 2011, 02:41

Stmt1: AD=2.5
In triangle ABD, tan <bad = 5/2.5=2
Hence we know the value of <bad
Also, AB^2=AD^2+BD^2
=2.5^2+5^2= 31.25
AB =\sqrt{31.25}
In triangle ABC, <ABC is right angle.
We know one side (AB) and one angel (<bad) in triangle ABC. Sufficient to find AC.

Stmt2: By similar reason, in triangle BDC we can find out <bcd
Also, BC^2=5^2+10^=125
BC = \sqrt{125}
In triangle ABC we know one side (BC) and one angel (<bcd). Sufficient to find AC.

OA D.
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Re: Length of AC [#permalink]

03 Nov 2011, 10:45

D. It can be visualized if you split the triangle into 2; then notice you have a proportion with 3 numbers missing 1.

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Re: Length of AC [#permalink]

14 Jan 2012, 16:37

I'm not able to understand the ans to this q.. could someone elaborate please?

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Re: Length of AC [#permalink]

14 Jan 2012, 17:27

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karthiksms wrote:

I'm not able to understand the ans to this q.. could someone elaborate please?

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In the diagram above, if arc ABC is a semicircle, what is the length of AC?

You should know the following properties to solve this question:
• A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle.

So, as given that AC is a diameter then angle ABC is a right angle.

• Perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.

Thus, the perpendicular BD divides right triangle ABC into two similar triangles ADB and BDC (which are also similar to big triangle ABC). Now, in these three triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles). For example: AB/AC=AD/AB=BD/BC. This property (sometimes along with Pythagoras) will give us the following: if we know ANY 2 values from AB, AD, AC, BC, BD, CD then we'll be able to find the other 4. We are given that BD=5 thus to find AC we need to know the length of any other line segment.

Also in such kind of triangles might be useful to equate the areas to find the length of some line segment, for example area of ABC=1/2*AC*BD=1/2*AB*BC (for more check: triangles-106177.html, geometry-problem-106009.html, mgmat-ds-help-94037.html, help-108776.html)

(1) AD = 2.5. Sufficient.
(2) DC = 10. Sufficient.

Answer: D.

Hope it helps.
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Re: Length of AC [#permalink] 14 Jan 2012, 17:27

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In the diagram above, if arc ABC is a semicircle, what is

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In the diagram above, if arc ABC is a semicircle, what is

In the diagram above, if arc ABC is a semicircle, what is

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