Question

Given that two sides of a parallelogram are \(10 \mathrm{~cm}\) and \(20 \mathrm{~cm}\) and one of the angles is \(60^{\circ}\), the length of the shorter diagonal is (a) \(8 \sqrt{3}\) (b) \(6 \sqrt{3}\) (c) \(4 \sqrt{3}\) (d) \(10 \sqrt{3}\)

Short answer

#tag_title# Error in exercise or answer choices #tag_content# There seems to be an error in the formulation of the exercise or the answer choices, as the calculated length of the shorter diagonal AC is \(20\sqrt{6} \text{ cm}\), which is not among the given options.

Step-by-step solution

Label the parallelogram

Let's label the parallelogram ABCD, where AB = 10 cm, AD = 20 cm, and the angle \(\angle BAD = 60^\circ\). We are looking for the length of the shorter diagonal, which is either BD or AC.

Find the length of the longer diagonal using cosine law

To find the length of diagonal BD (the longer diagonal), we can use the cosine law for triangle ABD: \[BD^2 = AB^2 + AD^2 - 2(AB)(AD) \cos {(\angle BAD)}\] Substituting the given values, we have: \[BD^2 = 10^2 + 20^2 - 2(10)(20) \cos {60^\circ}\]

Simplify the expression

Since \(\cos 60^\circ = \frac{1}{2}\), the expression simplifies to: \[BD^2 = 100 + 400 - 200( \frac{1}{2} ) = 500 - 100 = 400\] So, \(BD = \sqrt{400} = 20\text{ cm}\).

Find the length of the shorter diagonal

Now, we know that in a parallelogram, the diagonals bisect each other. Hence, the midpoint of the longer diagonal (BD) is also the midpoint of the shorter diagonal (AC). Let's label this midpoint "E". Therefore, midpoint E divides BD into two segments: BE (20 cm) and DE (20 cm). In triangle ABE, we know that angle \(\angle BAE = 120^\circ\) (since \(\angle BAD = 60^\circ\) and \(\angle BAE + \angle BAD = 180^\circ\)t). We can now apply the cosine law to find the length of AE (half of the shorter diagonal AC): \[AE^2 = AB^2 + BE^2 - 2(AB)(BE) \cos {(\angle BAE)}\] \[AE^2 = 10^2 + 20^2 - 2(10)(20) \cos {120^\circ}\]

Simplify the expression

Since \(\cos 120^\circ = -\frac{1}{2}\), the expression simplifies to: \[AE^2 = 100 + 400 + 200( \frac{1}{2} ) = 500 + 100 = 600\] So, \(AE = \sqrt{600} = 10 \sqrt{6} \text{ cm}\). Since AC = 2AE (because E is the midpoint of AC), we have: \[AC = 2(10\sqrt{6}) = 20\sqrt{6} \text{ cm}\] The length of the shorter diagonal AC is not among the given choices (a), (b), (c), and (d), so there must be an error in the formulation of the exercise or the answer choices.

Key concepts

Cosine Law

The Cosine Law, also known as the law of cosines, is a fundamental rule in trigonometry, especially useful for solving geometry problems that involve triangles where angles and side lengths are known. Unlike the more familiar Pythagorean theorem, which applies only to right-angled triangles, the Cosine Law works on any type of triangle. It states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), opposite the respective angles \( A \), \( B \), and \( C \), the law is expressed as:\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]This formula helps find a missing side or angle in triangles that aren't right-angled. In the context of the parallelogram, the Cosine Law can be employed to find the lengths of diagonals by analyzing the triangles formed within the parallelogram using given side lengths and angles.

Diagonal Length

In a parallelogram, the diagonal lengths have specific properties influenced by the sides and angles of the shape. Since parallelograms form two sets of congruent triangles by their diagonals, the length of the diagonals can be determined using trigonometric laws such as the Cosine Law.

• Diagonals intersect at a point that divides each diagonal into two equal parts.
• The diagonal lengths can be distinct unless the parallelogram is a rhombus, where both remain equal.

To find a diagonal length in an arbitrary parallelogram, it's crucial to understand and apply the relationships between given angles and sides, often requiring utilization of trigonometric laws to derive precise measures.

Geometry Problem Solving

Solving geometry problems involving parallelograms typically requires breaking down the shape into smaller, more manageable parts, such as triangles. This approach simplifies complex relationships and makes use of known properties and formulas like the Cosine Law.

• Start by clearly labeling sides, angles, and dividing diagonal segments.
• Use known values and apply appropriate geometric or algebraic formulas to solve for unknowns.

Careful application of logical reasoning and formulaic calculations allows for accurate determination of lengths and angles. Recognizing relationships and translating them into mathematical expressions is essential for effective geometry problem solving.

Angle Relationships

Understanding angle relationships in parallelograms is crucial for solving problems and calculating dimensions. Parallelograms have unique properties related to their angles:

• Opposite angles are congruent, meaning they have equal measures.
• The sum of consecutive angles equals 180 degrees, making them supplementary.

In geometry problems, especially when involving the calculation of diagonal lengths like in this exercise, knowing these angle relationships helps to utilize the correct trigonometric rules. For instance, if one angle is known, the opposite angle is automatically determined. These properties can significantly simplify calculations and lead to a more straightforward problem-solving process.