CAT EXAM #3 Math Question

[walter.hoffman]

Hi,

Can you please help with the following question? Why do we not need to calculate the actual squared numbers to verify the right triangle?


If the length of side AB is 17, is triangle ABC a right triangle?

(1) The length of side BC is 144.

(2) The length of side AC is 145.


Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

According to the Pythagorean Theorem, in a right triangle a2 + b2 = c2.

(1) INSUFFICIENT: With only two sides of the triangle, it is impossible to determine whether a2 + b2 = c2.

(2) INSUFFICIENT: With only two sides of the triangle, it is impossible to determine whether a2 + b2 = c2.

(1) AND (2) SUFFICIENT: With all three side lengths, we can determine if a2 + b2 = c2. It turns out that 172 + 1442 = 1452, so this is a right triangle. However, even if it were not a right triangle, this formula would still be sufficient, so it is unnecessary to finish the calculation.

The correct answer is C.

[RonPurewal]

Hi,

Can you please help with the following question? Why do we not need to calculate the actual squared numbers to verify the right triangle?

you know that the answer is going to turn out to be EXACTLY ONE of "yes" and "no"... and that's what "sufficient" means.

you don't care which one it is; if it's definitely yes or definitely no, then the statement is sufficient.

[shinpad86]

Can you help me understand why we can't use the statements on their own? If we know the length of AB is 17 and BC is 144 could we not calculated (17)squared + (144)squared = C squared and then add the two values on the left and take the square root of C and square root of the left side of the equation to determine that is is a right triangle?

[RonPurewal]

Can you help me understand why we can't use the statements on their own? If we know the length of AB is 17 and BC is 144 could we not calculated (17)squared + (144)squared = C squared and then add the two values on the left and take the square root of C and square root of the left side of the equation to determine that is is a right triangle?

you can't do the purple thing unless you ALREADY KNOW that the triangle is a right-angled triangle.

[RonPurewal]

you can also see the flaws in that reasoning by using (very elementary) visualization.

• take a stick.
• cut off one length that is 17 mm long.
• cut off another length that is 144 mm long.
• join them at one end.
• arrange them into various angles... anywhere from just above 0º to just below 180º.
• ALL of these arrangements will create triangles. (the third side connects the other ends of the sticks—the ends that are not stuck together.)
• the vast majority of these will not be right-angled triangles.

--

finally... even if this triangle were a right-angled triangle, why would you assume that AB and BC must be the legs?

even if this triangle IS a right-angled triangle, there are still two possibilities:
17^2 + (missing side)^2 = 144^2,
17^2 + 144^2 = (missing side)^2.