In the diagram above, triangle ABC has a right angle at B and a perime

Question

https://gmatclub.com/forum/download/file.php?id=68142 In the diagram above, triangle ABC has a right angle at B and a perimeter of 120. Line segment BD is perpendicular to AC and has a length of 24. AB > BC. What is the ratio of the area of triangle ABD to the area of triangle BDC? A. 9/7 B. 16/13 C. 16/11 D. 16/9 E. 16/7 Triangle.png

Aks111 · Answer

Was a bit tricky for me, tried to assume what the triangle could be and was able to find possible answer.

Since it is a right angled triangle assuming 3,4,5 as triplets of the triangle we find that a perimeter of 120 can be obtained by multiplying each side by 10.
So sides of triangle become AB =40; BC=30 (because AB>BC); and AC=50

Triangle ABD is similar to triangle BDC because BD is perpendicular to AC. We get,
AD*CD = BD^2 = 24^2
CD = AC-AD = 50-AD
AD*(50-AD) = 24^2
Let, AD=x
x^2 - 50x + 24^2 = 0
x^2 - 32x - 18x + 576 =0
(x-32)(x-18) = 0
=> AD= 32 or AD=18
Because AB>BC we have AD>CD 
=> AD=32 and CD=18

A(△ABD) / A(△BDC) = AD*BD / CD*BD = AD/CD = 32/18 = 16/9

Answer: D

wadhwakaran · Answer

Since, the perimeter of the triangle ABC is 120

AB=x, BC=y, AC=z and AB>BC (x>y)

BD=24

x+y+z=120

Pythagorean Triplet (3,4,5 has a sum of 12) meaning thereby that we can replace x,y,z by a multiple of this triplet.

x=40, y=30 and z=50

Now that Tr (ABD) is similar to Tr (BCD)

Ratio of Area of similar triangles = ratio of square of corresponding sides
Ar (ABD)/Ar (BCD)=AB*AB/BC*BC=40*40/30*30=16/9

Therefore option D is the answer.

chetan2u · Answer

Let us see what we are asked. 
Area of ABD= \frac{1}{2}*AD*BD 
Area of CBD= \frac{1}{2}*CD*BD 
Thus their ratio = Area of ABD : Area of CBD = \frac{1}{2}*AD*BD :  \frac{1}{2}*CD*BD =AD:CD
 So what we have in options is AD/CD

Next, triangles ABD and CBD, also triangle ABC, are similar, and we will get AD*CD=BD^2.
That is so because, triangles ABD and CBD give us  \frac{AD}{BD}=\frac{BD}{CD}

Thus,  AD*CD=BD^2=24^2

Various ways to solve further

1) AD and CD have to be factors of 24^2, so AD/CD too should have only factors of 24^2. ONLY 16/9 possible.

2) The ratio AD/CD should come out as some square/some square. Again 16/9, which is 4^2/3^2 fits in.

3) We can take the help of perimeter given to solve.

D

asterisk357 · Answer

in #3 how can we use the perimeter to get the ratio?

bumpbot · Answer

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In the diagram above, triangle ABC has a right angle at B and a perime

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