In a triangle ABC above, BD⊥AC. What is the value of b?

Question

Competition Mode Question https://gmatclub.com/forum/download/file.php?id=55194 In a triangle ABC above, BD⊥AC. What is the value of b? A. \sqrt{5} B. 12 C. 6\sqrt{5} D. 6\sqrt{6} E. \sqrt{18} 1.png

Babineaux · Accepted Answer

a^2 + b^2 = 18^2.........1

Again from ABD & BDC,

8^2 + BD^2 = b^2 ........ 2

And 10^2 + BD^2 = a^2 ........3

Putting 2&3 in equation 1, we obtain BD^2 = 80 .........4

Putting 4 in equation 2,

We get b=12

IMO ans is B

Posted from my mobile device

MayankSingh · Answer

IMO B

In Δ BDA & Δ CBA 
∠BAC=∠DAB (Common)
∠CBA=∠BDA = 90 
So both triangles are similar => sides proportional...

AD/AB = AB/AC
=> AB ^ 2 = AD x AC = 8 x 18
AB = √4 x 2 x 9 x 2 = 2x2x3=12

yashikaaggarwal · Answer

Using Pythagoras Theorem
=> AB^2+BC^2 = AC^2
=> B^2+A^2 = 18^2 ----------(1)
BD is perpendicular on AC.
Hence ABD and BCD are right angle triangle at D
∆ABD
=> BD^2+AD^2 = AB^2
=> BD^2+8^2 = B^2 
=> BD^2 = B^2-8^2 ----------(2)
∆BCD
=> BD^2+CD^2 = BC^2
=> BD^2+10^2 = A^2
=> BD^2 = A^2-10^2 ---------(3)
Comparing equation 2&3
=> A^2-10^2 = B^2-8^2
=> A^2-B^2 = 10^2-8^2
=> A^2-B^2 = 100-64
=> A^2-B^2 = 36 -----------(4)
Comparing equation 4&1
=> A^2+B^2 = 324
=> A^2-B^2 = 36
=> 2A^2 = 360
=> A^2 = 180
=> A = 6√5 --------(5)
Putting value in equation 1
=> A^2+B^2 = 324
=> 180+B^2 = 324
=> B^2 = 144
=> B = 12

Answer is B

Posted from my mobile device

Lipun · Answer

BD^2 = b^2 - 8^2 = a^2 - 10^2 
=>  a^2 - b^2 = 36

We know,  a^2 + b^2 = 18^2 = 324

=>  2b^2 = 288 => b = 12

Vinayak1996 · Answer

In right angled triangle ADB
8^2+BD^2=b^2

In right angled triangle BDC
10^2 + BD^2= a^2

Adding both the eqs and rearranging we get

164+2BD^2=a^+b^2
 
But a^2+b^2=18^2..from right angled triangle ABC

Therefore BD^2=80

and b^2=144
b=12

Posted from my mobile device

Saasingh · Answer

ANSWER = B

From the figure, we can say that:

BD² = AD x DC. ( i have learnt this fact for triangles in which "BD perpendicular AC" and "AB perpendicular BC" TO IMPROVE MY SPEED).

But just to prove above logic, We get this BD² = AD x DC because say 
Angle A = x. 
Angle ABD = 90-x
And thus angle DBC will be = x

From both small triangles, find tan x. 
In triangle ADB, tan x = BD/AD
In triangle DBC, tan x = CD/ BD 
---> BD² = AD*BD

Therefore, BD² = 80.

From traingle ABD, BD² + AD² = b²

---> b = \sqrt{80+64}

---> b = 12.

TheProfessor99 · Answer

DB^2 + DC^2 = BC^2  ==>  DB^2 = a^2 - 100  ---(1)
 DB^2 + AD^2 = AB^2  ==>  DB^2 = b^2 - 64  ---(2)

Also,
 AB^2 + BC^2 = AC^2  ---(3) ==>  a^2 = (18)^2 - b^2

From, 1 & 2
 a^2 - 100 = b^2 - 64  
 a^2 - b^2 = 36  ---(4)

From, 3 & 4
  (18)^2 - b^2 - b^2 = 36 
  (18)^2 - 2b^2 = 36 
  2b^2 = 324-36 
  2b^2 = 288 
  b^2 = 144 
  b = 12

Hence,  option B

Uapovsky · Answer

ADB and CDB both have 90 degrees since BD⊥AC.

We can make 2 equations based on the given data using Pythagorean theorem:

1) 8^2 + BD^2 = b^2 => BD^2 = b^2 - 64
2) 10^2 + BD^2 = a^2 => BD^2 = a^2 - 100

Combining 2 equations gives: 
3) b^2 - 64 = a^2 - 100 => a^2 - b^2 = 36 => a^2 = 36 + b^2

From the Triangle we also know that 
4) a^2 +b^2 = 18^2 => a^2 = 324 - b^2

from (3) and (4) we can solve:

36 + b^2 = 324 - b^2
 2b^2 = 288
 b^2 = 144
 b = 12

Answer: B

Satishqwerty · Answer

In triangle ABD and triangle ABC, 
 \angle BAD = \angle BAC  as it is the common angle. 
and 
 \angle ADB = \angle ABC  = 90 degrees
By AA similarity, 
Since, two angles are equal so the third angle will be equal as well. 
In this similarity, ratio of sides opposite to equal angles is equal.

So, the sides opposite  \angle BAD = \angle BAC  are BD and BC. 
The sides opposite  \angle ADB = \angle ABC  are AB and AC
and the sides opposite  \angle ABD = \angle ACB  are AD and AB.

Taking the ratio

\frac{BD}{BC} = \frac{AB}{AC} = \frac{AD}{AB}

\frac{BD}{a} = \frac{b}{18} = \frac{8}{b}

We need to find the value of b, so considering the pair -

\frac{b}{18} = \frac{8}{b}

Cross multiplying, we get

b^2 = 18 	imes 8

b^2 = 3^2 	imes 2^4

As side can only be negative, so will consider only the positive root.

b = 3 	imes 2^2

b = 12.

IMO,  OA, B

Rishikasaini2807 · Answer

12 is the answer... Pythagoras theorem

Posted from my mobile device

aadhithyak · Answer

Lets consider line BD to be x.

Given, ABC is a right triangle. So, a^2+b^2=18^2

As BD perpendicular to AC, ADB and BDC are right triangles. 
So, 8^2 + x^2 = b^2 and 10^2+x^2=a^2
 
Solving for b,
(10^2+x^2 )+ b^2= 18^2 
10^2+ (b^2-8^2) + b^2 = 18^2
2(b^2) = 288 
b^2 = 144
b = 12

Choice B

Posted from my mobile device

lacktutor · Answer

Triangle ABC: 
—>  a^{2} + b^{2} = 324

Since BD is a perpendicular to AC, 
—>  b^{2} —64 = a^{2} —100 
 a^{2} —b^{2} = 36

Adding together: 
—>  2a^{2} = 360  
 a^{2} = 180  
—>  b^{2} = 144  
 b = 12

Answer (B)

Posted from my mobile device

GMATWhizTeam · Answer

Solution 
 • In triangle ABC,  a^2 + b^2 = 18^2 ………Eq.(i) 
• In triangle, BDC,  a^2 – BD^2 = 10^2 …………Eq.(ii) 
• In triangle BDA,  b^2 -BD^2 = 8^2 …………Eq.(iii) 
• Subtracting Eq.(iii) from Eq.(ii), we get,
 o  a^2 -b^2 = 6^2 …………Eq.(iv)  
• Now, subtracting Eq.(iv) from Eq.(i), we get, 
 o  2b^2 = 18^2 – 6^2 ⟹ 2b^2 = 6^2 *8 ⟹ b = 12   
Thus, the correct answer is  Option B.

exc4libur · Answer

Two triangles with congruent angle and side, are similar.

b/8=18/b, b^2=144, b=12

Ans (B)

patrapan · Answer

Answer is 12 using the pythagorean theorem

HoneyLemon · Answer

Ans is B .

Please refer attached solution .

monikakumar · Answer

solution:
BA^2=AD*Ac
b^2=8*18
b^2=144
b=12

Ans B

OliveOyl2005 · Answer

Answer to this question is B.

using Pythagoras theorem in all 3 triangles individually, keeping BD common in the 2 smaller triangles, will help us get the answer.

sambitspm · Answer

Okay, on the first thought, I started by equating the area. Remind you that I was doing this calculation mentally to prepare for the Whiteboard. 
BUt I realized that the answer would then be dependent on a

a*b/2 = 18*sqrt(a^2+b^2)/2

Then I started with options. Clearly b^2 <324

This means we had options remaining A, B, E.
I started putting options and finding if they match the triangles and found out that    12    was the option 
 if b = 12, then BD = sqrt(80), then a would be sqrt(180). Now b^2 + a^2 = 324 = AC^2 
 
IMO B

In a triangle ABC above, BD⊥AC. What is the value of b?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

In a triangle ABC above, BD⊥AC. What is the value of b?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards