Are all angles of triangle ABC

[fiza gupta]

Are all angles of triangle ABC smaller than 90 degrees?
(1) 2AB = 3BC = 4AC
(2) AC^2 + AB^2 > BC^2

OA:A

[Jay@ManhattanReview]

Are all angles of triangle ABC smaller than 90 degrees?
(1) 2AB = 3BC = 4AC
(2) AC^2 + AB^2 > BC^2

OA:A

We have to see whether each of the three angles A, B and C are smaller than 90 deg. If any one angle is greater than 90, we have an answer as NO, else YES.

S1: 2AB = 3BC = 4AC imples than AB > BC > AC.

Say AB = 6, this mean that BC = 4 and AC = 3. Or AB : BC : AC :: 6 : 4 : 3

Let us recall the most popular triplet for a right angle traingle: 5 : 4 : 3. For the triple 5 : 4 : 3, the longest side is the hypotenuse and the angle opposite to it = 90. However, in our case, the ratio of the sides is 6 : 4 : 3.

We see that 6^ > 4^2 + 3^2. This implies that the trangle is an obtuse angle traingle (one angle > 90), which is at the opposite of the longest side.

=> All the angles are not smaller than 90. Answer is NO. A unique answer.

S2: Given that: AC^2 + AB^2 > BC^2.

If AB = AC = BC, each angle = 60. Answer is YES.

However, if AB = 6; AC = 3; BC = 4. angle C > 90 as seen in statement 1. Answer is NO. No unique answer. Insufficient.

Hope this helps!

-Jay
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[kevincanspain]

Wonderful analysis! Note that if you know the ratio of the lengths of the sides, you know that there is only one combination of angles possible, as all triangles with sides of a given ratio are similar to one another. Since there is complete certainty regarding the angles,we can say that (1) is sufficient without knowing whether the answer is yes or no!

[Jay@ManhattanReview]

Wonderful analysis! Note that if you know the ratio of the lengths of the sides, you know that there is only one combination of angles possible, as all triangles with sides of a given ratio are similar to one another. Since there is complete certainty regarding the angles,we can say that (1) is sufficient without knowing whether the answer is yes or no!

Thank you, Kevin.