Question

Find the maximum possible area of a triangle with sides of length 7 units and another side of length 8 units.
A). 24 sq. units
B). 28 sq. units
C). 30 sq. units
D). 27 sq. units

Answer 1

Hint: we have been given the sides of the triangle and length of the triangle (height) and we have been asked to find the area of the triangle. First, we need to know the formula of the area of triangle which is given by
\[A=\dfrac{1}{2}\times b\times h\times \sin (\alpha )\] Where (b) is the base of a triangle, (h) is the height of a triangle and \[\alpha \,\]is the angle between base and height According to the question we have to find maximum area that means \[\sin (\alpha )\,\]is maximum which means \[\sin (\alpha )=1\]and after substituting this value and further simplification we get the value of area.

Complete step-by-step solution:
We have given the sides of a triangle and height of a triangle
And we need to find the area of the triangle
So, we have to consider the sides of a triangle is (b)
Consider the height of a triangle is (h)

A triangle with two given sides has a maximum area if these two sides are placed at right angles to each other. For this triangle, one of the given sides can be considered as a base and other sides can be considered as height (because they meet at right angles).
Thus, plug these sides into the formula for the area of the triangle is given by
\[A=\dfrac{1}{2}\times b\times h\times \sin (\alpha )\]
Where, (b) is the base of a triangle, (h) is the height of a triangle and \[\alpha \,\]is the angle between base and height
According to the question it has mentioned the maximum possible area of a triangle that means \[\sin (\alpha )\,\] is maximum only when \[\sin (\alpha )\,=1\]so, we can say that \[\alpha ={{90}^{\circ }}\].
So, substituting the value \[\sin (\alpha )\,=1\] in the above formula we get:
 \[A=\dfrac{1}{2}\times b\times h\]
As we know that the value of \[b=7\] units and \[h=8\]units substitute this value on above equation we get:
\[A=\dfrac{1}{2}\times 7\times 8\]
After simplifying this above equation we get:
\[A=7\times 4\]
Further solving this we get:
\[A=28\] Sq. units
Therefore, the maximum possible area for triangle is 28 sq. units
So, the correct option is “option B”.

Note: In this problem it has been mentioned that we have to find the maximum possible area of a triangle that means two sides are met at right angles. Then it is said that two given sides have a maximum area. Hence, we can apply the formula for the area of the triangle.