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If AD is 6 and ADC is a right angle, what is the area of triangular re

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15 Oct 2017, 18:25
Bunuel wrote:

Given: $$AD=6$$. Question: $$area_{ABC}=\frac{1}{2}*AD*BC=\frac{1}{2}*6*(BD+DC)=3(BD+DC)=?$$

(1) Angle ABD = 60 --> triangle ABD is 30-60-90 triangle, so the sides are in ratio $$1:\sqrt{3}:2$$ --> as AD=6 (larger leg opposite 60 degrees angle) then $$BD=\frac{6}{\sqrt{3}}$$ (smallest leg opposite 30 degrees angle). But we still don't know DC. Not sufficient.

(2) AC=12, we can find DC. But we still don't know BD. Not sufficient.

(1)+(2) We know both BD and DC, hence we can find area. Sufficient.

Hi Bunuel

Cant we assume that that AD bisects BC into two?

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15 Oct 2017, 21:36
zanaik89 wrote:
Bunuel wrote:

Given: $$AD=6$$. Question: $$area_{ABC}=\frac{1}{2}*AD*BC=\frac{1}{2}*6*(BD+DC)=3(BD+DC)=?$$

(1) Angle ABD = 60 --> triangle ABD is 30-60-90 triangle, so the sides are in ratio $$1:\sqrt{3}:2$$ --> as AD=6 (larger leg opposite 60 degrees angle) then $$BD=\frac{6}{\sqrt{3}}$$ (smallest leg opposite 30 degrees angle). But we still don't know DC. Not sufficient.

(2) AC=12, we can find DC. But we still don't know BD. Not sufficient.

(1)+(2) We know both BD and DC, hence we can find area. Sufficient.

Hi Bunuel

Cant we assume that that AD bisects BC into two?

No. The height and the median coincide only in isosceles/equilateral triangle, when the height is dropped from the vertex formed by equal sides to the base.
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16 Oct 2017, 02:46
zanaik89 wrote:

Given: $$AD=6$$. Question: $$area_{ABC}=\frac{1}{2}*AD*BC=\frac{1}{2}*6*(BD+DC)=3(BD+DC)=?$$

(1) Angle ABD = 60 --> triangle ABD is 30-60-90 triangle, so the sides are in ratio $$1:\sqrt{3}:2$$ --> as AD=6 (larger leg opposite 60 degrees angle) then $$BD=\frac{6}{\sqrt{3}}$$ (smallest leg opposite 30 degrees angle). But we still don't know DC. Not sufficient.

(2) AC=12, we can find DC. But we still don't know BD. Not sufficient.

(1)+(2) We know both BD and DC, hence we can find area. Sufficient.

Hi Bunuel

Cant we assume that that AD bisects BC into two?

So u mean that since it isnt given that triangle ABC is an isoceles triangle or equilateral triangle, we can not consider AD as a perpendicular bisector right?
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Re: If AD is 6 and ADC is a right angle, what is the area of triangular re  [#permalink]

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16 Oct 2017, 02:53
zanaik89 wrote:

So u mean that since it isnt given that triangle ABC is an isoceles triangle or equilateral triangle, we can not consider AD as a perpendicular bisector right?

Yes, we know that AD is perpendicular to BC but we don't know whether it bisects BC.
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Re: If AD is 6 and ADC is a right angle, what is the area of triangular re  [#permalink]

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31 Aug 2018, 08:58
If AD is 6 and ADC is a right angle, what is the area of triangular region ABC?[/b]

Given: $$AD=6$$. Question: $$area_{ABC}=\frac{1}{2}*AD*BC=\frac{1}{2}*6*(BD+DC)=3(BD+DC)=?$$

(1) Angle ABD = 60 --> triangle ABD is 30-60-90 triangle, so the sides are in ratio $$1:\sqrt{3}:2$$ --> as AD=6 (larger leg opposite 60 degrees angle) then $$BD=\frac{6}{\sqrt{3}}$$ (smallest leg opposite 30 degrees angle). But we still don't know DC. Not sufficient.

(2) AC=12, we can find DC. But we still don't know BD. Not sufficient.

(1)+(2) We know both BD and DC, hence we can find the area. Sufficient.

Bunuel - My doubt is if a triangle like this which has a 30-60-90 angles, shouldn't the the triangle ADC should also be 30-60-90 making triangle ABC an equilateral triangle? because an equilateral triangle can be formed by joining two identical triangles, each with 30-60-90 angles.
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Re: If AD is 6 and ADC is a right angle, what is the area of triangular re  [#permalink]

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09 Jul 2019, 07:17
Bunuel VeritasKarishma The stem says that ADC is a right angle. That means AD is perpendicular to DC. Why are we assuming that B, D and C are in a straight line?
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Re: If AD is 6 and ADC is a right angle, what is the area of triangular re  [#permalink]

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10 Jul 2019, 06:21
GittinGud wrote:
Bunuel VeritasKarishma The stem says that ADC is a right angle. That means AD is perpendicular to DC. Why are we assuming that B, D and C are in a straight line?

BC is a straight line and D lies on it. Note sure why you think right triangle ADC messes that up.
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Re: If AD is 6 and ADC is a right angle, what is the area of triangular re  [#permalink]

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10 Jul 2019, 08:47
VeritasKarishma We have no information in the stem to decide whether BC is a straight line. Why are we assuming it is?
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Re: If AD is 6 and ADC is a right angle, what is the area of triangular re  [#permalink]

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10 Jul 2019, 11:12
AccipiterQ wrote:

I thought if you dropped a line down from a triangle vertex and it formed a right angle on the opposite side then that line bisected the side? So in this case if you know what BD is then you know what DC is?

To figure out whether it holds, why don't you try drawing some extreme figures, say, something like this:
Attachment:
Ques3.jpg

Will this be true in this case?
When will it be true? When the triangle is equilateral, sure. Also when the triangle is isosceles if the equal sides form the angle from which the altitude is dropped.

Don't put your faith in the figure given. It may be just one of the many possibilities or may be somewhat misleading.

Hi! How do we know that the ADC is also a 90 30 60 triangle?
Re: If AD is 6 and ADC is a right angle, what is the area of triangular re   [#permalink] 10 Jul 2019, 11:12

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