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Length of the leg of right triangle ABC

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by gmattesttaker2 » Mon Mar 17, 2014 9:39 pm
Hello,

For the following:

Right triangle ABC's hypotenuse measures 20 inches. If AB and BC are its
legs, what is the length of AB?

(1) AB - 4 = BC.
(2) The ratio of AB to BC is 4:3.

OA: D


I was trying to solve as follows:

AB^2 + BC^2 = AC^2

1) AB - 4 = BC
=> ( 4 + BC )^2 + BC^2 = AC^2
=> ( 4 + BC )^2 + BC^2 = 20^2

Now upon solving this equation I got:
( BC + 16 ) ( BC - 12 ) = 0

=> BC = 12. Once I know BC I can solve for AB.

My question is if I have such a quadratic equation do I need to solve it in a DS question or can I safely assume that only 1 value will be positive and therefore conclude it to be Sufficient?


2) (4x)^2 + (3x)^2 = 20^2
=> 16x^2 + 9x^2 = 400
=> 25x^2 = 400
=> x^2 = 16
=> x = 4

Hence, AB = 4(4) = 16 - Suff.

Hence D


Thanks for your help,
Sri
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by Matt@VeritasPrep » Tue Mar 18, 2014 12:08 am
gmattesttaker2 wrote:My question is if I have such a quadratic equation do I need to solve it in a DS question or can I safely assume that only 1 value will be positive and therefore conclude it to be Sufficient?
It depends on the question. On a Euclidean geometry problem, no: the positive solution will be the only solution, as a polygon can't have sides of negative length. On a pure algebra question, yes: for instance, I could ask you if x > -19, then say that x² - 30x + 221 = 0. You'd certainly need to check the solutions to be sure.

As for this question, I'd solve it this way. Let AB = a and BC = b. We know a² + b² = 400, and we want the value of a.

S1 tells us that b = (a + 4), so we have a² + (a+4)² = 400, or a² + a² + 8a + 16 = 400, or a² + 4a - 192 = 0. This will have one negative and one positive solution, so the positive solution must be the only valid one, sufficient.

S2 tells us that it's a 3:4:5 triangle (remember your Euclidean triples!) Thus we don't have to do any algebra at all: the ratio of the three sides is set, so we only need one side (which we have, the hypotenuse) to solve, sufficient.
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by GMATGuruNY » Tue Mar 18, 2014 6:11 am
gmattesttaker2 wrote: Right triangle ABC's hypotenuse measures 20 inches. If AB and BC are its
legs, what is the length of AB?

(1) AB - 4 = BC.
(2) The ratio of AB to BC is 4:3.
Start with the easier statement.
Statement 2 suggests that ∆ABC is a 3:4:5 triangle.

Statement 2: AB:BC = 4:3
In other words, BC:AB = 3:4.
Since 3:4:5 = 12:16:20, statement 2 implies that BC=12, AB=16, and AC=20.
SUFFICIENT.

Statement 1: AB - 4 = BC
Notice that the values implied by statement 2 -- BC=12, AB=16, and AC=20 -- also satisfy statement 1.
No other combination for AB and BC will yield a hypotenuse of 20.
If the values for AB and BC decrease, than AC will be less than 20.
If the values for AB and BC increase, then AC will be greater than 20.
Thus, like statement 2, statement 1 implies that AB=16.
SUFFICIENT.

The correct answer is D.
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by ceilidh.erickson » Tue Mar 18, 2014 2:11 pm
Whenever you see a right triangle in a coordinate plane (or really, a right triangle anywhere), the vast majority of the time it will be one of the SPECIAL right triangles that you should know:
special side lengths: 3-4-5, 5-12-13, 8-15-17
special angles: 45-45-90, 30-60-90

In other words... you should almost NEVER do Pythagorean theorem!

The fact that the hypotenuse of the triangle is 20 strongly suggests that it's probably a 12-16-20 triangle (a multiple of 3-4-5).

The question is asking us for the length of AB. The only unknowns at this point are AB and BC, since the hypotenuse is given. We already know that AB^2 + BC^2 = 20^2, so all we need is one more relationship between them to have a system of equations. So your rephrased question is:
Can we define AB in terms of BC?

(1) AB - 4 = BC.
If you have one of the legs defined in terms of the other, and you know the hypotenuse, you would then only have one unknown: AB. Don't actually do the work - it has to be solvable! Even though it would give us a quadratic equation, we know that there can only be 1 solution in a geometry problem. Sufficient.

(2) The ratio of AB to BC is 4:3.
Once again, this defines AB in terms of BC. This also proves our suspicion that it's a 12-16-20 triangle. Sufficient.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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