Triangle PQR Solving For Angles, Area, And Side Length Conditions
- QR = 3 units
- PR = x units
- PQ = 2x units
- Angle PQR = θ
This article delves into the geometric properties of triangle PQR, focusing on determining the cosine of angle θ, calculating θ and the area of the triangle when x = 2.4 units, and establishing the conditions for the triangle's existence. We will use the Law of Cosines and the formula for the area of a triangle to solve these problems.
7.1 Showing that cos θ = (x² + 3) / (4x)
To demonstrate that cos θ = (x² + 3) / (4x), we will employ the Law of Cosines. The Law of Cosines is a fundamental trigonometric principle that relates the side lengths of a triangle to the cosine of one of its angles. In triangle PQR, the Law of Cosines can be expressed as follows:
PR² = PQ² + QR² - 2(PQ)(QR)cos θ
Substituting the given values, we have:
x² = (2x)² + 3² - 2(2x)(3)cos θ
Now, we simplify the equation step by step:
x² = 4x² + 9 - 12x cos θ
Our goal is to isolate cos θ on one side of the equation. To do this, we first move the terms involving x² to the left side:
12x cos θ = 4x² + 9 - x²
Combining like terms, we get:
12x cos θ = 3x² + 9
Finally, we divide both sides by 12x to solve for cos θ:
cos θ = (3x² + 9) / (12x)
We can simplify this expression further by factoring out a 3 from the numerator:
cos θ = 3(x² + 3) / (12x)
Now, we can cancel the common factor of 3:
cos θ = (x² + 3) / (4x)
Thus, we have successfully shown that cos θ = (x² + 3) / (4x). This equation provides a crucial link between the side length x and the cosine of angle θ, which will be instrumental in our subsequent calculations. This relationship highlights the power of the Law of Cosines in solving geometric problems involving triangles with known side lengths and angles. The ability to express cos θ in terms of x allows us to further analyze the triangle's properties and solve for unknown values, such as the angle θ itself and the area of the triangle. This is a key step in understanding the geometry of triangle PQR and its various characteristics.
7.2 If x = 2.4 units:
7.2.1 Calculate θ
Now that we have the expression for cos θ in terms of x, we can calculate the value of θ when x = 2.4 units. We substitute x = 2.4 into the equation we derived earlier:
cos θ = ((2.4)² + 3) / (4 * 2.4)
First, we calculate (2.4)²:
(2.4)² = 5.76
Now, substitute this value back into the equation:
cos θ = (5.76 + 3) / (9.6)
Simplify the numerator:
cos θ = 8.76 / 9.6
Divide to find the value of cos θ:
cos θ ≈ 0.9125
Now, we need to find the angle θ whose cosine is approximately 0.9125. To do this, we use the inverse cosine function (also known as arccos or cos⁻¹):
θ = arccos(0.9125)
Using a calculator, we find:
θ ≈ 24.14 degrees
Therefore, when x = 2.4 units, the angle θ in triangle PQR is approximately 24.14 degrees. This calculation demonstrates how the relationship between the side length x and cos θ, derived from the Law of Cosines, can be used to determine specific angles within the triangle. Understanding the angle θ is crucial for further analysis of the triangle's properties, such as its area and the relationships between its sides and angles. This step is essential in providing a complete picture of the triangle's geometry and its characteristics when x is given as 2.4 units.
7.2.2 Calculate the area of triangle PQR
To calculate the area of triangle PQR, we can use the formula:
Area = (1/2) * PQ * QR * sin θ
We are given that QR = 3 units and PQ = 2x units. Since x = 2.4 units, we have:
PQ = 2 * 2.4 = 4.8 units
We have already calculated θ to be approximately 24.14 degrees. Now, we need to find the sine of θ:
sin(24.14°) ≈ 0.4087
Now, we can substitute these values into the area formula:
Area = (1/2) * 4.8 * 3 * 0.4087
Multiply the values:
Area ≈ (1/2) * 14.4 * 0.4087
Area ≈ 7.2 * 0.4087
Area ≈ 2.94 square units
Thus, the area of triangle PQR when x = 2.4 units is approximately 2.94 square units. This calculation demonstrates the application of the formula for the area of a triangle using two sides and the included angle. The area provides another key piece of information about the triangle's size and shape, complementing our knowledge of its side lengths and angles. The ability to calculate the area is essential in various geometric applications and helps us to fully characterize the triangle's properties. This result, combined with the previously calculated angle θ, provides a comprehensive understanding of triangle PQR when x is 2.4 units.
Calculate the values of x for which the triangle exists
For a triangle to exist, the triangle inequality theorem must hold true. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In triangle PQR, this gives us three inequalities:
- PQ + QR > PR
- PR + QR > PQ
- PQ + PR > QR
Substituting the given values, we have:
- 2x + 3 > x
- x + 3 > 2x
- 2x + x > 3
Let's solve each inequality:
- 2x + 3 > x
Subtract x from both sides:
x + 3 > 0
Subtract 3 from both sides:
x > -3
Since x represents a side length, it must be positive, so this inequality tells us that x > 0.
- x + 3 > 2x
Subtract x from both sides:
3 > x
This inequality can be rewritten as:
x < 3
- 2x + x > 3
Combine like terms:
3x > 3
Divide both sides by 3:
x > 1
Now, we need to consider all three inequalities together:
- x > 0 (from inequality 1)
- x < 3 (from inequality 2)
- x > 1 (from inequality 3)
Combining these inequalities, we find that the values of x for which the triangle exists are:
1 < x < 3
This range of values ensures that the triangle inequality theorem is satisfied, and therefore, a valid triangle can be formed with the given side lengths. This condition is crucial for the geometric feasibility of the triangle and highlights the importance of the triangle inequality theorem in determining the validity of triangle constructions. Understanding this range of possible x values is essential for further analysis and application of triangle PQR in various contexts.
