Problem 21 - Algebra 2 and Trigonometry

Question

A kite is in the shape of a quadrilateral with two pair of congruent adjacent sides. The lengths of two sides are 20.0 inches and the lengths of the other two sides are 35.0 inches. The two shorter sides meet at an angle of \(115^{\circ} .\) a. Find the length of the diagonal between the points at which the unequal sides meet. Write the length to the nearest tenth of an inch. b. Using the answer to part a, find, to the nearest degree, the measure of the angle at which the two longer sides meet.

Step-by-Step Solution

Answer

a. 47.0 inches; b. 12 degrees.

Step 1: Understand the Problem

We have a kite (a quadrilateral) with two pairs of adjacent congruent sides: one pair is 20 inches, and another pair is 35 inches. The angle between the two shorter sides is given as \(115^{\circ}\). We need to find the length of the diagonal between the points where 20-inch and 35-inch sides meet and determine the angle between the longer sides.

Step 2: Apply the Law of Cosines for Part a

To find the diagonal between the points where the unequal sides meet (20-inch and 35-inch), we use the Law of Cosines: \[ d^2 = a^2 + b^2 - 2ab\cos(C) \] where \(a = 20\ inches\), \(b = 35\ inches\), and \(C = 115^{\circ}\). Substitute in the values: \[ d^2 = 20^2 + 35^2 - 2 \times 20 \times 35 \times \cos(115^{\circ}) \]. Then solve for \(d\).

Step 3: Calculate and Simplify for Part a

Calculate \(d^2 = 400 + 1225 - 2 \times 20 \times 35 \times \cos(115^{\circ})\). The cosine of 115° is approximately -0.4226, so \[ d^2 = 400 + 1225 + 588.28 \]. Then calculate \(d^2 = 2213.28\) and \(d \approx \sqrt{2213.28} \approx 47.0\ inches\). The length of the diagonal to the nearest tenth is \(47.0\) inches.

Step 4: Understand Part b

Now that we have found the length of the diagonal, we can find the angle between the two longer sides (the 35-inch sides). Let's label the angle \( \theta\). We will again use the Law of Cosines in one of the triangles formed by the diagonal, the 20-inch and the 35-inch sides.

Step 5: Use the Law of Cosines for Part b

This time, use the triangle formed by two sides of 35 inches and the diagonal we found (47 inches). Use the Law of Cosines: \[ c^2 = a^2 + b^2 - 2ab\cos(\theta) \] when \(a = 35\ inches\), \(b = 35\ inches\), and \(c = 47\ inches\). Substitute the values to solve for \(\theta\): \[ 47^2 = 35^2 + 35^2 - 2 \times 35 \times 35 \times \cos(\theta) \].

Step 6: Calculate and Solve for Part b

Calculate: \(2209 = 1225 + 1225 - 2 \times 35 \times 35 \times \cos(\theta)\). Simplify to \(2209 = 2450 - 2450\cos(\theta)\). Rearrange to find \(\cos(\theta)\): \(245\cos(\theta) = 241\) leads to \(\cos(\theta) = \frac{241}{245}\).Finally, solve for \(\theta\) using the inverse cosine function \( \theta \approx \cos^{-1}\left(\frac{241}{245}\right) \approx 11.8^{\circ}\). The measure of the angle to the nearest degree is \(12^{\circ}\).

Key Concepts

Law of Cosines

The Law of Cosines is a powerful tool in geometry, especially when dealing with non-right triangles. It connects the lengths of the sides of a triangle with the cosine of one of its angles. The formula is given by:

• \( c^2 = a^2 + b^2 - 2ab\cos(C) \)

Here, \(c\) is the side opposite angle \(C\), while \(a\) and \(b\) are the other two sides of the triangle.
This law becomes incredibly useful in kite geometry when you need to find unknown side lengths or angles in a kite, quite like our problem. By knowing two sides and the included angle, you can determine the third side using this law, making it easy to solve problems involving kite diagonals or angles.

Quadrilateral Properties

A kite is a special type of quadrilateral. Quadrilaterals have four sides and, in the case of a kite, the sides are grouped into two pairs of adjacent congruent sides. This properties set kites apart from other quadrilaterals:

• The diagonals are perpendicular to each other.
• One of the diagonals bisects the other.
• The adjacent sides are congruent in pairs.

Understanding these properties is crucial because they determine how you can apply the laws of trigonometry and geometry when calculating distances and angles, as demonstrated by solving the diagonal and angle measurement problems in our given kite example.

Angle Measurement

Measuring angles correctly in kites is essential, particularly since they can vary quite a bit across different parts of the kite. The properties of a kite ensure that

• the angles between unequal sides are supplementary when connected by the bisecting diagonal,
• special tools like the Law of Cosines or the Law of Sines can be utilized to calculate unknown angles.

In our specific task, the angle between the shorter sides was provided (\(115^{\circ}\)), allowing us to find the length of applicable diagonals. Then, using the newly found diagonal, we determined the angle between the longer sides with the information calculated from hooks like cosine inverse functions.

Diagonals in Kites

Diagonals in kites serve as a cornerstone for working with their properties. They offer interesting interactions such as

• partitioning the kite into two congruent triangles,
• intersecting perpendicularly, and
• in more cases, one serves as the symmetry line or bisector.

This unique characteristic helps in calculating unknown lengths or properties through simple mathematical approaches.
By calculating one diagonal using the given sides and angle, and using trigonometric principles, we also inferred the angles connecting the longer sides, revealing vital properties that solve the problems within kite geometry.