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705-805 (Hard)|   Geometry|      
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Bunuel
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Triangle ABC has a right angle at B. Point D is the foot of the altitu 28 Mar 2019, 05:55
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Triangle ABC has a right angle at B. Point D is the foot of the altitude from B, AD = 3, and DC = 4. What is the area of triangle ABC ?


(A) \(4\sqrt{3}\)

(B) \(7\sqrt{3}\)

(C) 21

(D) \(14\sqrt{3}\)

(E) 42

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Triangle ABC has a right angle at B. Point D is the foot of the altitu 28 Mar 2019, 19:48
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Bunuel

Triangle ABC has a right angle at B. Point D is the foot of the altitude from B, AD = 3, and DC = 4. What is the area of triangle ABC ?


(A) \(4\sqrt{3}\)

(B) \(7\sqrt{3}\)

(C) 21

(D) \(14\sqrt{3}\)

(E) 42

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Angle ACD=Angle BAD, Because if Angle c=x, angle a= 90.-x, in Triangle ABD, Angle ABD=180-(90+90-x)
=x, Similarly, Angle BCD= Angle ABD,

Using Similarlity,

Let AD=x,

\(\frac{4}{x}=\frac{x}{3}\)

\(x=2\sqrt{3}\)

\(Area=1/2*2\sqrt{3}*(4+3)=7\sqrt{3}\)

IMO, Option B
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Triangle ABC has a right angle at B. Point D is the foot of the altitu 16 Oct 2019, 09:45
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chetan2u Can you help with this one?

I believe Ypsychotic is saying CD/BD=BD/AD. From this we can find BD and with that we can find area.

Can you share your approach?
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Triangle ABC has a right angle at B. Point D is the foot of the altitu 16 Oct 2019, 10:56
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chetan2u Can you help with this one?

I believe Ypsychotic is saying CD/BD=BD/AD. From this we can find BD and with that we can find area.

Can you share your approach?
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Hi,

Yes similar triangle property can be used to find BF and as you too have mentioned that BD =\(\sqrt{AD*CD}=\sqrt{12}\) and I too will do it same way.

But even the choices give away the answer and we should be able to exploit this too.
The area is \(\frac{1}{2}*7*BD\).
So our answer should normally be multiple of 7.
A is out.
Now take C. If (1/2)*7*BD=21...BD=6
But is it possible .. NO as in triangle aBDC, BC will become >7, that is a side of a right angled triangle will become bigger than the hypotenuse AC or 3+4.
So answer is less than 21 for sure. Only B left.
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Triangle ABC has a right angle at B. Point D is the foot of the altitu 20 Oct 2023, 07:57
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