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# In the figure above, if AC = CD, then r =

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Math Expert
Joined: 02 Sep 2009
Posts: 58335
In the figure above, if AC = CD, then r =  [#permalink]

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26 Feb 2019, 01:13
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Difficulty:

15% (low)

Question Stats:

92% (00:59) correct 8% (01:43) wrong based on 38 sessions

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In the figure above, if AC = CD, then r =

A. 45 – s
B. 90 – s
C. s
D. 45 + s
E. 60 + s

Attachment:

2019-02-26_1208_001.png [ 15.21 KiB | Viewed 450 times ]

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Re: In the figure above, if AC = CD, then r =  [#permalink]

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26 Feb 2019, 02:19
isosceles triangle so ; s+r = 45
angle r = 45-s
IMO A
Bunuel wrote:

In the figure above, if AC = CD, then r =

A. 45 – s
B. 90 – s
C. s
D. 45 + s
E. 60 + s

Attachment:
2019-02-26_1208_001.png

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Re: In the figure above, if AC = CD, then r =  [#permalink]

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26 Feb 2019, 03:04
Bunuel wrote:

In the figure above, if AC = CD, then r =

A. 45 – s
B. 90 – s
C. s
D. 45 + s
E. 60 + s

Attachment:
2019-02-26_1208_001.png

c = 90

thus, angle A = Angle D.

180 - 90 = 90

90/2 = 45 for angle A and Angle D.

r + s = 45

r = 45 - s.

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Re: In the figure above, if AC = CD, then r =  [#permalink]

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26 Feb 2019, 05:26
it should be "A" as ACD is an isosceles triangle.
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Re: In the figure above, if AC = CD, then r =  [#permalink]

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26 Feb 2019, 05:44
It should be A if the given angle is actually 90.

( it is not given in question, we assumed it )

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Re: In the figure above, if AC = CD, then r =  [#permalink]

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27 Feb 2019, 02:13
Top Contributor
Hello,
Greetings for the day!

This is a fairly easy question based on the properties of a right angled triangle and also on the basic properties of triangles.

1. The sum of the interior angles in ANY triangle is always equal to 180 degrees.
2. If two sides of a triangle are equal, the corresponding angles opposite to these, will also be equal.
3. A right angled triangle is one which contains ONE right angle; because of this, the other two angles will be acute angles.

The right angled triangle given in the question is a special case of a right angled triangle – it is an Isosceles right angled triangle since two of its perpendicular sides are equal i.e. AC = CD. Therefore, the angles opposite to these sides are also equal i.e. angle CAD = angle CDA.
Also, since angle ACD = 90 degrees, angle CAD + angle CDA = (180 – 90) = 90 degrees.
Hence, angle CAD = angle CDA = (90/2) = 45 degrees.
But, angle CDA = r + s; therefore, r + s = 45.
Hence, r = 45 – s.
The same is illustrated in the diagram shown below.

Attachment:

In the figure above, if AC = CD, then r =.jpg [ 29.92 KiB | Viewed 240 times ]

If you have worked out enough problems in Geometry and built up sufficient muscle memory, problems like these can be taken down in about 30 seconds, because application of some of the basic concepts mentioned above will happen at a subconscious level. The only stage where you have to consciously put in an effort is to understand that the triangle is an isosceles right angled triangle. Also, in such easy problems, make sure that you do not relax while marking the right option, lest you end up marking the wrong option in the excitement of having solved an easy question.
Hope this helps.
Cheers!
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Re: In the figure above, if AC = CD, then r =  [#permalink]

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27 Feb 2019, 02:55
Isosceles triangle hence the total angle must be 45 degrees

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Re: In the figure above, if AC = CD, then r =   [#permalink] 27 Feb 2019, 02:55
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