# Geometry Triangles Data Sufficiency

Here is a DS question on Geometry on basic properties of triangles.
Is triangle ABC with sides a, b and c acute angled?
1. Triangle with sides a^2, b^2, c^2 has an area of 140 sq cms.
2. Median AD to side BC is equal to altitude AE to side BC.

The correct answer is Choice A. Statement 1 alone is sufficient, while statement 2 is not.

Explanation

From the information given we have to determine if the given triangle is an acute angled triangle.

The information given to us is the measure of the sides of the triangle.

If a, b and c measure the sides of a triangle, and let us say 'a' is the longest side of the triangle, then
1. the triangle is acute angled if a^2 < b^2 + c^2
2. right angled if a^2 = b^2 + c^2 and
3. obtuse angled if a^2 > b^2 + c^2

Now let us evaluate the statements given to us

Statement 1: Triangle with sides a^2, b^2, c^2 has an area of 140 sq cms.
The statement provides us with one valuable information. We can form a triangle with sides a^2, b^2, c^2.

For any triangle we know that sum of two sides is greater than the third side.
So,  a^2 < b^2 + c^2

That is the condition to be satisfied for a triangle with sides a, b and c to be an acute angled triangle.
Statement 1 answers in the positive and the information is sufficient.

Statement 2: Median AD to side BC is equal to altitude AE to side BC
We can infer that the triangle is either equilateral or isosceles. An equilateral triangle is definitely an acute angled triangle. However, an isosceles triangle need not be an acute angled triangle.

So, statement 2 is not sufficient.

Statement 1 is sufficient, while statement 2 is not sufficient. Choice A is the correct answer.

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