What is x in the diagram below?

MANHATTAN GMAT OFFICIAL SOLUTION:

You can calculate the area of the triangle, using the side of length 12 as the base:
(1/2)(12)(3) = 18

Next, use the side of length 7 as the base and write the equation for the area:
(1/2)(7)(x) = 18

Now solve for x, the unknown height:
7x = 36
x = 36/7

Answer: C.

Hi Bunel,
Could you kindly explain the highlighted portion in detail?
I am not able to understand why you have equated that to 18?
Thanks,
Arun

In the given triangle with solid lines, the area of the triangle = 0.5*base*height.

Now you need to understand that the height can be from any of the 3 vertices and the base will then be the side to which the height is perpendicular to.

In this case, the line with length 3 is the height for the base with length 12, giving you area of the triangle = 0.5*12*3 = 18

Similarly, for the same triangle, the area will also be = 0.5*7*x ...( as the side with x length is perpendicular to the 'base' with length 7 units).

For a particular triangle, the area MUST be the same irrespective of what base or height combination you choose.

Thus , you equate, 0.5*7*x = 18 ---> x = 36/7.

Hope this helps.

I was trying to solve it with the solid line's length=7, will it be more clearly marked in the exam? After going through the solution i realized that they meant the Solid+dotted line length is equal to 7

Hi Bunuel,

I am confused about what the 7 is referring to. Is 7 the solid line? If so, how are we using that 7 as a base of the imaginary triangle drawn by the dotted lines? Also, why are we able to reference the area of 18 from the triangle with solid lines to help us determine the base of the dotted line triangle? This one is driving me crazy. Thank you in advance for your reply!

Wow... thank you Bunuel! Makes much more sense.

I solved it this way...

Triangle DBA is Similar to triangle CEA
So, \(\frac{3}{x} = \frac{7}{12}\), \(x = \frac{36}{7}.\)

I added some letters to help guide the solution.



Area of triangle = (1/2)(base)(height)
IMPORTANT CONCEPT: we can use ANY of the three sides as our base.

So, for example, if we want to find the area of triangle ABC, we can use side AB as the base, or we can use side AC as the base, or we can use side BC as the base.

If we use side AB as the base, then the base has length 12 and the height is 3
So, area of triangle ABC = (1/2)(12)(3)

If we use side AC as the base, then the base has length 7 and the height is x
So, area of triangle ABC = (1/2)(7)(x)

IMPORTANT: If we use side AB as the base, the area of the triangle will be the same as the area we get if we use side AC as the base.

So, (1/2)(12)(3) = (1/2)(7)(x) [solve for x]
Divide both sides by 1/2 to get: (12)(3) = (7)(x)
Divide both sides by 7 to get: 36/7 = x

Answer: C

Cheers,
Brent

OA: C

Sin A =\(\frac{3}{7}=\frac{x}{12}\)
x=\(\frac{36}{7}\)

Sent from my XT1068 using GMAT Club Forum mobile app

A student asked whether we can use similar triangles to answer this question. So.....

First add some letters to the diagram to get:


At this point, we can see that we have two similar triangles:

That is, ∆ACE is similar to ∆ADB

We can see that side AE corresponds to side AB (since both are hypotenuses)
And we can see that side CE corresponds to side BD (since both sides are opposite the angle denoted by the circle)

Key concept: With any two SIMILAR triangles, the ratios of corresponding sides will be equal.
We can write: AE/AB = CE/BD
Replace with actual lengths: 12/7 = x/3
Cross multiply to get: 7x = 36
Solve: x = 36/7

Answer: C

Cheers,
Brent

We can start by focusing on similar triangles when there are no obvious lengths to evaluate. Note using the labeling in the attachment, ACE and ABD are similar triangles. This means we have this nice ratio of AC/CE = AB/BD. Then BD = 12*3/7 = 36/7. The answer is C.

Hi BrentGMATPrepNow, If we use side AC as the base in triangle ABC, not sure why the height is not 3 here but x outside of triangle ABC? As thought x is the height for triangle e.g AEB? Then dotted line CE part will be missing as base if AC/E is use as the base for triangle AEB? Have I missed something here? Thanks Brent

The height of a triangle is always perpendicular to the triangle's base.
If we choose side AC as the base, DC (with length 3) is perpendicular to the base AC, whereas BE (with length 5) is NOT perpendicular to the base AC.

Understand now thanks BrentGMATPrepNow. height CD is perpendicular to base AB and BE(x) is perpendicular to base AC.

You're all making this too complicated.

Problem solving questions are, by default, drawn to scale.

x is clearly bigger than 3, and smaller than 7. It appears to be something like halfway between them. This fits answer C.