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I am having problem with the following question:

From the triangle which is greater Area of the Triangle ABC or 27 enter image description here

Now how would I find the height of such a triangle.

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You do not need to. Note the height is less than 9. – David Mitra Jul 10 '12 at 22:39

From the information given, it is not possible to determine the height of the triangle. However, you can still answer the question.

Taking $6$ as the base of the triangle, you know that the height must be less than or equal to $9$. It is less than $9$, if the triangle is not right. So the area of the triangle is less than or equal to $\frac{1}{2}(9)(6) = \frac{1}{2}(54) = 27$. Hence the area of the triangle is less than or equal to $27$. If the triangle is not right, then the area is less than 27.

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Were the angle at $B$ right, the triangle would have an area of 27. That gives the triangle the maximum height. Hence, the area of the triangle is less than 27 as drawn, since the angle at $B$ is nonright.

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