Question

What is the largest area of an isosceles triangle with two edges of length 3?

Answer 1

Hint: For solving this question you should know about the isosceles triangle. And you should also know how we calculate the area of this triangle. The isosceles triangle is a triangle which has two sides of equal length. And the angles via these sides on the other side are always equal. And the area of this is the half of the multiplication of base and height.

Complete step by step solution:
According to the question we have to calculate the largest area of an isosceles triangle whose two sides are of length 3.
If we see the diagram of an isosceles then it is clear that in this triangle two sides are equal to each other and the angle made by these sides with third side are always equal.

The area of this triangle is equal to the \[\dfrac{1}{2}\times \] Base \[\times \] Height which is \[\dfrac{1}{2}\times BC\times h\].
And here, \[h=\sqrt{A{{B}^{2}}-{{\left( \dfrac{BC}{2} \right)}^{2}}}\]
So, if we come to our question, then here the height is considered as h.

Here, the \[BC=2BD\].
And \[BD=\sqrt{{{3}^{2}}-{{h}^{2}}}\]
So, the area of the isosceles triangle is
\[A=\dfrac{1}{2}\times h\times \left( 2\times \sqrt{{{3}^{2}}-{{h}^{2}}} \right)=h\sqrt{9-{{h}^{2}}}\]
So, the Area is a geometric mean of two positive quantities \[{{h}^{2}}\] and \[9-{{h}^{2}}\]. Which gives an arithmetic mean (as 4.5).
Since, the geometric mean of any quantity can not be larger than their arithmetic mean. So, the largest possible value for the area is 4.5.
The maximum area is achieved when \[h=\sqrt{4.5}\].
OR if we see it by \[\sin \theta \]
Then the area is given by \[\dfrac{1}{2}\times 3\times 3\times \sin \theta \], where \[\theta \] is the vertex angle of the isosceles triangle. This has also the maximum value of 4.5 at \[\theta ={{90}^{\circ }}\].
So, the largest possible area is 4.5 units\[^{2}\].

Note: During calculating the area for a triangle you have to take all the measurements in the same unit and if any measurement is not given then you can find them by Pythagoras theorem and trigonometric identities. And make sure to accurately calculate the calculations unless your answer will be wrong.