Problem 16 - Algebra 2 and Trigonometry

Question

Find the exact value of the area of an isosceles triangle if the measure of a leg is 12 centimeters and the measure of the vertex angle is 45 degrees.

Step-by-Step Solution

Answer

The area of the triangle is \( 36\sqrt{2} \) square centimeters.

Step 1: Understand the Triangle

An isosceles triangle has two equal sides, which are called the legs, and the vertex angle is the angle between these two legs. We need to calculate the area of this triangle with the given measurements.

Step 2: Set Up the Formula for Area

The formula for the area of a triangle is \( A = \frac{1}{2}ab\sin C \), where \( a \) and \( b \) are the lengths of the legs, and \( C \) is the vertex angle.

Step 3: Plug in the Given Values

In this problem, both sides \( a \) and \( b \) are equal to 12 cm, and the angle \( C \) is 45 degrees. Substitute these values into the formula: \[ A = \frac{1}{2} \times 12 \times 12 \times \sin(45^{\circ}) \]

Step 4: Calculate \( \sin 45^{\circ} \)

The sine of 45 degrees is \( \frac{\sqrt{2}}{2} \). Use this to continue solving the expression for area.

Step 5: Solve for the Area

Substitute \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \) into the area formula: \[ A = \frac{1}{2} \times 12 \times 12 \times \frac{\sqrt{2}}{2} \]. Calculate: \( A = 72 \times \frac{\sqrt{2}}{2} \). This simplifies to \( A = 36\sqrt{2} \) square centimeters.

Key Concepts

Isosceles Triangle

An isosceles triangle is a special type of triangle with two sides of equal length. These equal sides are called the "legs" of the triangle. The third side, which can be of different length, is often referred to as the "base." The angle between the two equal sides is known as the "vertex angle." Because of the symmetry, the other two angles in an isosceles triangle, often called the base angles, are also equal. This property makes isosceles triangles unique and useful in many mathematical contexts.

When you encounter a problem involving isosceles triangles, it's advantageous to use their properties to simplify calculations or deduce unknown values. Recognizing the balance and equality in an isosceles triangle can help in forming equations or decisions regarding the geometry of the shape.

Vertex Angle

In any isosceles triangle, the vertex angle is the angle formed between the two equal sides, otherwise known as the legs. It acts as a pivotal point for understanding the triangle’s geometry. The vertex angle can tell us a lot about the shape and size of the triangle.

• If the vertex angle is very small, the triangle appears more elongated.
• If the vertex angle increases towards 90 degrees, the triangle becomes more symmetrical and equilateral-like.

This angle has practical significance in various calculations, particularly when applying trigonometric functions like sine, as it does in solving for the area of a triangle using the sine function.

Sine Function

The sine function is a fundamental part of trigonometry, often abbreviated as "sin." It relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. For angles such as 45 degrees, the sine value is exact and known, simplifying computations significantly.

In our context, knowing that \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \) is crucial for solving the problem. This function is invaluable in calculating not only angles but also areas, as it allows us to transform geometric side lengths and angles into tangible values contributing to shape area computations or other geometric properties.

Area of a Triangle

The area of a triangle represents the region enclosed within its three sides. It can be calculated using various methods, with one common formula involving the sine of an angle: \( A = \frac{1}{2}ab\sin C \), where \( a \) and \( b \) are sides of the triangle and \( C \) is the included angle.

This formula is particularly efficient for triangles where we know two side lengths and the included angle. By applying this, we reduce complex, multi-step processes to a single calculation, offering a quick route to finding the desired area. In the case of isosceles triangles, this formula can integrate identical leg measurements easily, multiplying the sine of the vertex angle to produce the area instantly. Understanding and using this formula can open up a quick way of understanding extensive geometric properties without repetitive calculations.