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805+ (Hard)|   Geometry|      
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Bunuel
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In triangle ABC above, if AD = BC ,what is the value of x? 27 Jan 2021, 01:15
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In triangle ABC above, if AD=BC ,what is the value of x?

(1) D is the midpoint of side AC.
(2) The angle ABC is a right angle.


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D
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In triangle ABC above, if AD = BC ,what is the value of x? 27 Jan 2021, 04:44
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Attachment:
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Angle BAD = 2x-x=x

AD=BC=BD.....(1)

Statement1: AD=DC...(2)

From (1) and (2)

DC=BC=BD or BCD is an equilateral triangle

2x=60 or x=30

Sufficient

Statement 2- Angle ABC=90

Since BD=BC, Angle BCD=2x

Angle DBC = 180-4x

Angle ABC = 180-3x = 90

x=30

Sufficient
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In triangle ABC above, if AD = BC ,what is the value of x? 27 Jan 2021, 15:16
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In triangle ABC above, if AD=BC ,what is the value of x?

Now, BD^2 = AD*DC

Stat1: D is the midpoint of side AC.
So, BD^2 = AD*DC or, BD^2 = DC^2 or, BD=DC=BC
So, 6x = 180 or, x= 30. Sufficient

Stat2: The angle ABC is a right angle.
BD^2 =BC^2 = CD*CA (By golden triangle)
Now, in triangle ABC, 2x+x+90 = 180 so, x = 30. Sufficient

So, I think D. :)
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In triangle ABC above, if AD = BC ,what is the value of x? 27 Jan 2021, 17:50
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nick1816
Attachment:
triangle.png

Angle BAD = 2x-x=x

AD=BC=BD.....(1)

Statement1: AD=DC...(2)

From (1) and (2)

DC=BC=BD or BCD is an equilateral triangle

2x=60 or x=30

Sufficient

Statement 2- Angle ABC=90

Since BD=BC, Angle BCD=2x

Angle DBC = 180-4x

Angle ABC = 180-3x = 90

x=30

Sufficient
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nick1816 Can you please explain how you came to the conclusion BD = AD?
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In triangle ABC above, if AD = BC ,what is the value of x? 27 Jan 2021, 23:53
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In triangle ABC above, if AD=BC ,what is the value of x?
ADB = 180-2x
So, BAD = 180 - [(180-2x) - x] = x
So, AD = BD

(1) D is the midpoint of side AC.
Therefore, AD = DC = BD = BC
So, BDC is an equilateral triangle.
All angels are 2x
2x = 60
x = 30
Sufficient

(2) The angle ABC is a right angle.
CBD = 90 - x
Also, BD = CD
So, BCD = 2x

In triangle BCD
2x + 2x + (90 - x) = 180
x = 30
Sufficient

Option D

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In triangle ABC above, if AD = BC ,what is the value of x? 28 Jan 2021, 03:24
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The sides opposite of equal angles are always equal.

dc2880
nick1816
Attachment:
triangle.png

Angle BAD = 2x-x=x

AD=BC=BD.....(1)

Statement1: AD=DC...(2)

From (1) and (2)

DC=BC=BD or BCD is an equilateral triangle

2x=60 or x=30

Sufficient

Statement 2- Angle ABC=90

Since BD=BC, Angle BCD=2x

Angle DBC = 180-4x

Angle ABC = 180-3x = 90

x=30

Sufficient

nick1816 Can you please explain how you came to the conclusion BD = AD?
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In triangle ABC above, if AD = BC ,what is the value of x? 28 Jan 2021, 09:25
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Bunuel

In triangle ABC above, if AD=BC ,what is the value of x?

(1) D is the midpoint of side AC.
(2) The angle ABC is a right angle.
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Great question!!

Important: Even though we aren't provided any angles, we can still label many of the angles in the diagram.

Since angles on a line must add to 180°, ∠ADB = (180 - 2x)°


Since angles in ∆ADB must add to 180°, ∠BAD = x°


Since ∆ADB has two equal angles (of x°), the triangle is an isosceles triangle which means side BD = side AD


Since side BD = side BC, we have an isosceles triangle which means ∠DCB = 2x°, and since all angles must add to 180 degrees, we can see that ∠BDC = (180 - 4x)°


Now that we filled in so much information, the two statements should be relatively easy to analyze....

Target question: What is the value of x?

Statement 1: D is the midpoint of side AC.
This means side DC = side AD

At this point we can see that ∆BDC is an equilateral triangle, which means each angle must be 60°
This means 2x° = 60°, which means x = 30
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: ∠ABC is a right angle.

Since the angles in ∆ABC must add to 180 degrees, we can conclude that ∠A + ∠C = 90°
In other words, x° + 2x° = 90°
Simplify: 3x° = 90°, which means x = 30
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent
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In triangle ABC above, if AD = BC ,what is the value of x? 22 Aug 2022, 18:07
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