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In the xy-plane, is the triangle that connects the 3 different points

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Math Revolution GMAT Instructor
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In the xy-plane, is the triangle that connects the 3 different points  [#permalink]

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New post 15 Jul 2018, 06:58
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A
B
C
D
E

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  75% (hard)

Question Stats:

35% (02:11) correct 65% (01:48) wrong based on 23 sessions

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[GMAT math practice question]

In the xy-plane, is the triangle that connects the 3 different points \(A(3,4), B(p,q)\), and \(C(r,s)\) a right triangle?

1) \((p-3)(r-3)+(q-4)(s-4)=0\)
2) \(p=3\) and \(s=4\)

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Re: In the xy-plane, is the triangle that connects the 3 different points  [#permalink]

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New post 15 Jul 2018, 07:34
1
MathRevolution wrote:
[GMAT math practice question]

In the xy-plane, is the triangle that connects the 3 different points \(A(3,4), B(p,q)\), and \(C(r,s)\) a right triangle?

1) \((p-3)(r-3)+(q-4)(s-4)=0\)
2) \(p=3\) and \(s=4\)


Nice question!!

Here is my approach for this problem.

Stat 1: Here lets try to reduce the the equation in slope form(We know for perpendicular lines product of slope is -1)
So (p-3)(r-3)+(q-4)(s-4)=0
(p-3)(r-3)=-(q-4)(s-4)=0
(p-3)/(q-4)=-(r-3)/(s-4)

Now vertex of triangle is given as A(3,4),B(p,q) and C(r,s)
So clearly from above equation we see lines AB and AC are perpendicular.

Stat2:

P=3 implies point B lies in the horizaontal line that passes through point A for which also X coordinate is 3.
s=4 implies point C lies in the vertical line that passes through point A for which also Y coordinate is 4.
So both lines will be perpindicular at A and hence a right angle triangle.

Option D is correct choice.

Press Kudos if it helps!!
Math Revolution GMAT Instructor
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Re: In the xy-plane, is the triangle that connects the 3 different points  [#permalink]

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New post 15 Jul 2018, 18:16
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Condition 2):
If \(p = 3\), then \(q ≠ 4\) since \((p,q)\) is different from \((3,4).\)
If \(s = 4\), then \(r ≠ 3\) since \((r,s)\) is different from \((3,4)\).
So, \((p,q)\) is on the line \(x = 3\) and \((r,s)\) is on the line \(y = 4.\)
As these lines are perpendicular AB and AC are perpendicular, and triangle ABC is a right triangle.
Thus, condition 2) is sufficient.

Condition 1) is complicated. If you can’t figure out how to apply condition 1), CMT4(B) tells you to choose D as the answer.

Condition 1)
\((p-3)(r-3)+(q-4)(s-4)=0\)
\(=> (p-3)(r-3) = -(q-4)(s-4)\)
\(=> \frac{(p-3)(r-3)}{(q-4)(s-4)} = -1\) or \((q-4)(s-4) = 0\)
\(=> {\frac{(p-3)}{(q-4)}} {\frac{(r-3)}{(s-4)}} = -1\) or \(q = 4\) or \(s = 4\)

Case 1: \(\frac{(p-3)}{(q-4)} * {\frac{(r-3)}{(s-4})} = -1\)
\(\frac{(p-3)}{(q-4)}\) and \(\frac{(r-3)}{(s-4)}\) are the slopes of two sides of the triangle.
Since the product of these slopes is \(-1\), the two sides are perpendicular and the triangle is a right triangle.

Case 2: \(q = 4\)
If \(q = 4\), then \(p\) is not \(3\). Also, we must have \((p-3)(r- 3) = 0\). So, \(r = 3\) since \((p,q)\) is different from \((3,4)\).
Thus, \((p,q)\) lies on the line \(y = 4\) and \((r,s)\) lies on the line \(x = 3\).
As these lines are perpendicular, AB and AC are perpendicular, and triangle ABC is a right triangle.

Case 3: \(s = 4.\)
A similar argument to the one used for case 2 shows that triangle ABC is a right triangle.

Thus, condition 1) is also sufficient.

Therefore, D is the answer.

Answer: D

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
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Re: In the xy-plane, is the triangle that connects the 3 different points &nbs [#permalink] 15 Jul 2018, 18:16
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