In triangle XYZ, side XY, which runs perpendicular to side YZ, measure

In triangle \(XYZ\), \(XY\) is perpendicular to \(YZ\); triangle is right triangle
\(XY\) = \(24\)
Longest side (\(XZ\)) = \(26\)
\(Area\) \(=\,\frac{1}{2} * XY * YZ\)
\(=\,\frac{1}{2}*24*\sqrt{26^2-24^2}\)
\(=\,\frac{1}{2}*24*\sqrt{100}\)
\(=\,120\)

Answer B

Hi All,

GMAT questions are often built around established "patterns" in the realm of Quant and Verbal. Learning to spot those patterns can save you time (and in the case of certain Quant questions, save you from having to do extra calculations).

From the given prompt, we know that we're dealing with a right triangle that has sides of 24 and 26 (AND that the 26 is the hypotenuse!). While you could use the Pythagorean Theorem to figure out the missing side, you might recognize that this is a 5/12/13 triangle that has been "doubled":

5/12/13
10/24/26

With this shortcut, you can quickly determine that the missing side is 10.

The question asks for the area of the triangle:

Area = (1/2)(Base)(Height) = (1/2)(10)(24) = 120

Final Answer:

GMAT assassins aren't born, they're made,
Rich

VERITAS PREP OFFICIAL SOLUTION:

Those employing Pythagorean Theorem are in for a fight, calculating a^2 + 24^2 = 26^2, then finding the length of a and calculating the area. But those who know the trusty 5-12-13 triplet can quickly see that if 24 = 12*2 and 26 = 13*2, then the other short side is 5*2 which is 10, and the area then is 1/2 * 10 * 24, which is 120. Knowing these ratios, this is a 30 second problem; without them it could be a slog of over 2 minutes, easily, with a higher degree of difficulty due to the extensive calculations. So on today of all days, Friday, the 5th day of the 12th month, keep that 13th in there as a lucky charm.

Would it be correct to take 1/2 the base *24 to get 12b, at which point one would realize that the answer choice has to be divisible by 12, therefore 120, answer choice B?

Hi BuggerinOn,

In this question, you CAN use that logic to your advantage and it would get you the correct answer. In other questions though, you would have to be careful. The prompt does NOT state the side lengths are integers, so on more complex versions of this question (in which the base is NOT an integer), your logic would NOT hold up and you might select an incorrect answer (and not even know it).

GMAT assassins aren't born, they're made,
Rich

Since XY is perpendicular to YZ, the triangle is a right triangle, and hence we can use the Pythagorean theorem to determine side YZ:

(XY)^2 + (YZ)^2 = (XZ)^2

24^2 + s^2 = 26^2

576 + s^2 = 676

s^2 = 100

s = 10

Since the area of a right triangle is half the product of the two legs, we have:

Area of triangle XYZ = ½(24)(10) = 120

Answer: B

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