Problem 33 - Algebra 2

Question

Find the area of \(\triangle A B C\) . Round your answer to the nearest tenth. $$ m \angle B=87^{\circ}, a=10.1, c=9.8 $$

Step-by-Step Solution

Answer

The area of the triangle is approximately 49.9 square units.

Step 1: Identify the given values

The problem provides values for angle B and sides a and c of the triangle. \nGiven: \nm \(\angle B = 87^{\circ}\), \na=10.1, \nc=9.8

Step 2: Apply the formula

The area (\(A\)) of a triangle with sides a and b, and with an included angle C is given by the formula \(A = 0.5ab\sin(C)\). Here, \(a=10.1, b=9.8\), and \(C=87^{\circ}\). So substitute these values into the formula.

Step 3: Calculate the area

After substituting the given values into the formula, use a calculator to find the area, rounding to the nearest tenth.

Key Concepts

Angle Measurement

Understanding angle measurement is essential when working with triangles. Angles in a triangle are measured in degrees, and the sum of the internal angles in any triangle adds up to 180 degrees. This foundational concept helps in understanding triangles' properties and relationships between their angles and sides.

In our problem, angle B is given as 87 degrees. This tells us that it is a large angle, just shy of a right angle, which is 90 degrees. Knowing the measurement of this angle helps us in calculating other parameters of the triangle, such as the area. Accurate angle measurement is crucial for precision in triangle calculations.

Sine Rule

The sine rule is a powerful tool in triangle geometry, especially when dealing with non-right triangles. It helps in finding unknown angles or sides when some parts are already known. The sine rule states:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

• "\(a, b, c\)" are the sides of the triangle.
• "\(A, B, C\)" are the opposite angles.

This rule shows the relationship between the sides of a triangle and its angles through trigonometric functions. However, in our exercise, we are using the sine of angle B directly in the area formula to simplify the area calculation rather than using it to find unknown parts.

Triangle Geometry

Triangle geometry involves various properties and formulas that help describe and solve triangle problems. One of the basic formulas for finding the area of a triangle involves using an angle and two sides. Known as the sine area formula, it is expressed as:\[A = 0.5 \, a \, b \, \sin(C)\]where "\(a\)" and "\(b\)" are the lengths of two sides, and "\(C\)" is the included angle.

This formula is especially useful for triangles that are not right-angled. In our given problem, plugging in the known side lengths and angle measurement allows us to directly calculate the triangle's area efficiently.

Trigonometry in Triangles

Trigonometry plays a critical role in solving problems involving triangles. By using trigonometric functions like sine, cosine, and tangent, we can derive relationships and formulas to simplify complex geometric problems. For example, when trying to find the area of a triangle, the sine function is particularly useful when you know two sides and the included angle.

In the given exercise, by applying the formula \(A = 0.5 \, a \, b \, \sin(C)\), we harness the power of trigonometry to find the area swiftly and accurately. Calculating \(\sin(87^\circ)\) and then following through with the multiplication provides a reliable way to determine the solution. Being familiar with these trigonometric principles enhances our understanding and ability to solve diverse problems in triangle geometry.