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Chapter 6: Problem 25

What is the largest possible area for a triangle that has one side of length 4 and one side of length \(7 ?\)

Short Answer

Expert verified
The largest possible area for a triangle with sides of length 4 and 7 is 14 square units, which occurs when the angle between these sides is \(90^\circ\), making the triangle a right-angled triangle.

Step by step solution

01

Identify the variables

The given sides of the triangle are length 4 and 7, say \(a=4\) and \(b=7\), respectively. Let \(\theta\) be the angle between these sides and c be the third side. To find the maximum area, we need to find the optimal angle \(\theta\).
02

Area of the triangle

To find the area A of the triangle, we use the formula involving the sine of the angle between the given sides: \[A = \frac{1}{2}ab\sin\theta\] In our case, this becomes: \[A = \frac{1}{2}(4)(7)\sin\theta\] So, \(A = 14\sin\theta\).
03

Using sine rule to find angle

We will now apply the sine rule to find a relationship between the angle \(\theta\) and its opposite side c. The sine rule states: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\] In our case, we know that \(c = 4\), so we have: \[\frac{7}{\sin\theta} = \frac{4}{\sin(180^\circ-\theta)}\]
04

Find the optimal angle

Our goal is to find the value of \(\theta\) such that the area A is maximized. By using properties of sine function, we know that: \[\sin\theta = \sin(180^\circ-\theta)\] The function sine takes its maximum value when its argument is 90 degrees. Thus, to maximize area A, we should choose \(\theta = 90^\circ\), which would make the triangle a right-angled triangle.
05

Calculate the maximum area

Using the optimal angle \(\theta = 90^\circ\) in the triangle area formula, we get: \[A = 14 \sin(90^\circ) = 14\] Thus, the largest possible area for the triangle is 14 square units.

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