fiza gupta wrote: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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