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In ∆ ABC above, which of the following could be a value of x?

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In ∆ ABC above, which of the following could be a value of x? [#permalink]

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In ∆ ABC above, which of the following could be a value of x?

(A) 12
(B) 8
(C) 4
(D) 2
(E) 1


[Reveal] Spoiler:
Attachment:
2017-08-24_1250_002.png
2017-08-24_1250_002.png [ 6.83 KiB | Viewed 468 times ]
[Reveal] Spoiler: OA

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Re: In ∆ ABC above, which of the following could be a value of x? [#permalink]

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New post 24 Aug 2017, 01:30
As per diagram
Angle A+C =90 degree
So A+(C/ something) < 90
Angle B has to be more than 90 to make triangle ABC =180
Option: A

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Re: In ∆ ABC above, which of the following could be a value of x? [#permalink]

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New post 24 Aug 2017, 01:41
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Bunuel wrote:
Image
In ∆ ABC above, which of the following could be a value of x?

(A) 12
(B) 8
(C) 4
(D) 2
(E) 1


[Reveal] Spoiler:
Attachment:
2017-08-24_1250_002.png

10x = sum of interior opposite angles which is greater than 90

only possible value is 120
A
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Re: In ∆ ABC above, which of the following could be a value of x? [#permalink]

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New post 29 Aug 2017, 16:00
Bunuel wrote:
Image
In ∆ ABC above, which of the following could be a value of x?

(A) 12
(B) 8
(C) 4
(D) 2
(E) 1


We can let the vertex of the right angle be D. Thus, triangle BCD is a right triangle with right angle at D. So, angle ABC (with measure 10x) is an exterior angle of triangle BCD and is equal to the sum of angle BDC and angle BCD (recall that an exterior angle of a triangle is equal to the two remote interior angles of the triangle). Since angle BDC is a right angle (90 degrees), angle ABC must be more than 90 degrees. From the answer choices, we see that only choice A (12) would make angle ABC more than 90 degrees.

Alternate Solution:

We can express the angle CBD (where D denotes the vertex of the right triangle) as 180 - 10x.

In a right triangle, the right angle is the greatest; therefore, we must have 180 - 10x < 90, or equivalently, 18 - x < 9. This inequality can be further simplified as x > 9. Only choice A (12) satisfies this requirement.

Answer: A
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Re: In ∆ ABC above, which of the following could be a value of x?   [#permalink] 29 Aug 2017, 16:00
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In ∆ ABC above, which of the following could be a value of x?

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