Statement 1:jain2016 wrote:In the figure, angle C and angle M are right angles, and KL = 10. If the area of triangle ABC is four times the area of triangle KLM, what is the length AB?
(1) Angles ABC and KLM have the same measure.
(2) LM is 6 inches.
OAA
Since ∠C=∠M=90 and ∠ABC=∠KLM, the remaining two angles are also equal:
∠BAC=∠LKM.
Triangles with the same combination of angles are SIMILAR.
In similar triangles, corresponding sides in the SAME RATIO.
Here, AB and KL are corresponding sides, since each is a hypotenuse.
A rule for two similar triangles:
If each side in the larger triangle is x times the corresponding side in the smaller triangle, then the area of the larger triangle is x² the area of the smaller triangle.
In the case at hand:
Since the area of ∆ABC is 4 times the area of ∆KLM, each side in ∆ABC is 2 times the corresponding side in ∆KLM.
Since AB and KL are corresponding sides, we get:
AB = 2(KL) = 2*10 = 20.
SUFFICIENT.
Statement 2:
Thus, ∆KLM is a 6-8-10 triangle, implying that the area of ∆KLM = (1/2)(6)(8) = 24.
Since the area of ∆ABC is 4 times the area of ∆KLM, the area of ∆ABC = 4*24 = 96.
Implication:
(1/2)(AC)(BC) = 96
(AC)(BC) = 192.
It's possible that AC=1 and BC=192, in which case AB² = 1² + 192².
It's possible that AC=12 and BC=16, in which case AB² = 12² + 16².
Since AB can be different values, INSUFFICIENT.
The correct answer is A.
