Problem 17 - Algebra 2 and Trigonometry

Question

Emily wants to draw a parallelogram with the measure of one side 12 centimeters, the measure of one diagonal 10 centimeters and the measure of one angle 120 degrees. Is this possible? Explain why or why not.

Step-by-Step Solution

Answer

No, it's not possible; the given dimensions and angle contradict the properties of a parallelogram.

Step 1: Understanding the Given Parallelogram

Emily wants to draw a parallelogram with a side measuring 12 cm, a diagonal measuring 10 cm, and an internal angle of 120 degrees. In a parallelogram, opposite angles are equal and consecutive angles are supplementary (sum to 180 degrees).

Step 2: Analyzing the Parallelogram Properties

For a parallelogram, the diagonal divides it into two congruent triangles. In our scenario, the diagonal of 10 cm must satisfy the law of cosines in one of these triangles. Let's consider one such triangle with sides 12 cm (side of the parallelogram), 10 cm (diagonal), and the included angle of 120 degrees that Emily has specified.

Step 3: Applying the Law of Cosines

The law of cosines states that for a triangle with sides \(a\), \(b\), and \(c\), and an angle \(C\) opposite to side \(c\), the following holds: \[c^2 = a^2 + b^2 - 2ab\cos(C)\]In this case, let the diagonal be denoted as \(c = 10\) cm, one side as \(a = 12\) cm, and the angle \(C = 120^{\circ}\):\[10^2 = 12^2 + b^2 - 2 \times 12 \times b \times \cos(120^{\circ})\]\(\cos(120^{\circ}) = -0.5\), so:\[100 = 144 + b^2 + 12b\]

Step 4: Simplifying and Solving for the Side

Rearrange the equation from the previous step to solve for \(b^2\):\[b^2 + 12b = 100 - 144 = -44\]This equation implies \(b^2 + 12b = -44\), which is impossible, as \(b^2\) and \(12b\) result in a positive and neutral value respectively, making them unable to sum to a negative number.

Step 5: Concluding the Feasibility

The above computation reveals a contradiction because the sum of a square and a positive term cannot equal a negative number. Therefore, it's not possible to form a parallelogram with the specified dimensions and angle without violating geometric principles.

Key Concepts

Understanding the Law of Cosines

The Law of Cosines is a fundamental tool in solving triangles. It's specifically useful when you have a triangle without a right angle, as it works in any triangle. The law states that in a triangle with sides labeled as \(a\), \(b\), and \(c\), and an angle \(C\) which is opposite to side \(c\), the relationship is given by:

• \(c^2 = a^2 + b^2 - 2ab \cos(C)\)

This formula connects the length of a side of a triangle to the other two sides and the cosine of the included angle. It's like a sibling to the Pythagorean Theorem, which only applies to right-angled triangles. Using this law, we can find an unknown side of a triangle if we know two sides and the included angle. In Emily's problem, applying this law was crucial to analyze if such a parallelogram could exist with given measurements.

Comprehending Triangle Congruence

Triangle congruence deals with the idea that if two triangles are congruent, they have the same size and shape, meaning their corresponding sides and angles are equal. There are specific ways to prove triangle congruence:

• Side-Side-Side (SSS)
• Side-Angle-Side (SAS)
• Angle-Side-Angle (ASA)
• Angle-Angle-Side (AAS)
• Right angle-Hypotenuse-Side (RHS) for right triangles

In the context of parallelograms, diagonals bisect the shape into two congruent triangles. However, if one tries to break down Emily's parallelogram using triangles, the mismatch in the side measurements, angle, and the diagonal's role reveals that these supposed triangles can't be congruent, hampering the formation of such a parallelogram.

Exploring Geometric Principles

Geometric principles provide the foundation for understanding shapes and their properties. A parallelogram has specific properties: opposite sides are equal, opposite angles are equal, and its diagonals bisect each other. These principles help us decipher whether given measurements can create specific shapes.
In the exercise, Emily is dealing with a highly constrained situation: creating a parallelogram from specific side lengths, angle, and diagonal. Known properties inform us that logically fitting the pieces requires satisfying certain numeric conditions, like those presented by the Law of Cosines. When such conditions result in mathematical contradictions, such as a negative sum involving squared numbers, it signals a violation of these geometric principles and the impossibility of the structure in question.