Problem 31 - Algebra 2

Question

Find the area of \(\triangle A B C\) . Round your answer to the nearest tenth. $$ m \angle A=52^{\circ}, a=9.71, c=9.33 $$

Step-by-Step Solution

Answer

The area of \(\triangle A B C\) when rounded off to the nearest tenth, gives the desired answer.

Step 1: Assigning Values

First, assign the given values to their corresponding variables. We have angle A which equals 52°, side a = 9.71 units and another side of triangle c = 9.33 units. In this particular case we denote side 'a' as 'b' and side 'c' as 'a' for our formula to be valid.

Step 2: Insert values into the formula

Next, insert the values obtained from Step 1 into the formula for the area of a triangle. Here, A denotes the angle, 'a' denotes the side opposite to measured angle and 'b' denotes the other given side \[\text{Area}=\frac{1}{2}ab\sin(A)\]So, substituting we get \[\text{Area}=\frac{1}{2}*9.33*9.71*\sin(52°)\]

Step 3: Computation

Now, calculate the value of the expression you've just written to find the area of the triangle. Use a calculator to compute the value or if you're doing it manually, first calculate the product 9.33*9.71 and then multiply the result by \(\sin(52°)\). After that, divide this product by 2.

Step 4: Rounding Off

The final step is to round off the answer to the nearest tenth as per the problem's requirement.

Key Concepts

Triangle Area Calculation

Calculating the area of a triangle can be very straightforward once you understand the formula. To find the area of a triangle when you have two sides and the included angle, you can use the formula: \[\text{Area} = \frac{1}{2} \cdot ab \cdot \sin(A)\]where \(a\) and \(b\) are the lengths of the two sides, and \(A\) is the included angle between them. This is particularly useful for non-right triangles, which don't have the simple base times height method.
One key step in using this formula is ensuring that the angle is between the two sides you are using. This formula provides a way to combine basic trigonometry with geometric understanding to provide accurate area measurements.

• Step 1: Assign the given values properly – match sides with their corresponding angles.
• Step 2: Use a calculator for trigonometry functions like sine to find their exact values.

This method is very practical when dealing with any arbitrary triangle and it's often favored in surveys or navigation. Once you perform the calculations, make sure to round off to the desired precision.

Sine Rule

The Sine Rule helps solve triangles when you have angles and sides that aren't directly linked by right angles. It's expressed as:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]This rule is advantageous when you have:

• Two angles and one side
• Two sides and a non-included angle

However, in our current context of calculating the area of a triangle, we use the sine of an angle to simplify area calculations. It's important to note the difference between using the Sine Rule for sides and angles versus using sine for area calculations. The idea is that the sine of an angle can relate the ratio of its opposite side, similar to trigonometric functions in right triangles.
Employing the sine function within trigonometry can extend your toolkit, especially for solving non-right angled triangles where conventional methods like the Pythagorean theorem fall short.

Mathematical Problem Solving

Mathematical problem solving involves a planned and logical approach to finding solutions. In our exercise, finding the area of a triangle required several key skills in problem-solving.

• Understand the problem: Identify what is given and what needs to be found.
• Develop a plan: Choose the right formula or approach that addresses the problem specifics, like using the triangle area formula when angles and non-adjacent sides are given.
• Carry out the plan: Substitute the values into the formula or equation correctly and calculate accurately using tools like calculators for precision.

It’s crucial to follow logical steps like rechecking your computation and ensuring all units and calculations align correctly. Once computed, adjusting your final answer to the required precision (like rounding off to the nearest tenth) ensures that the answer is presented properly.
By mastering such mathematical problem-solving techniques, you can easily tackle similar problems, improving both understanding and efficiency.