Problem 89 - Algebra and Trigonometry

Question

Dimensions of a Lot A parcel of land is 6 \(\mathrm{ft}\) longer than it is wide. Each diagonal from one corner to the opposite corner is 174 \(\mathrm{ft}\) long. What are the dimensions of the parcel?

Step-by-Step Solution

Answer

The width is 120 feet and the length is 126 feet.

Step 1: Define Variables

Let the width of the parcel be \( w \) feet. Then, the length of the parcel is \( w + 6 \) feet, since it is 6 feet longer than its width.

Step 2: Set Up the Equation Using the Diagonal

The diagonal of a rectangle creates a right triangle with the width and length as the other two sides. According to the Pythagorean theorem, it holds that: \[ w^2 + (w + 6)^2 = 174^2 \].

Step 3: Expand the Equation

Expand \((w + 6)^2\):\[ (w+6)^2 = w^2 + 12w + 36 \].Substitute back into the equation:\[ w^2 + w^2 + 12w + 36 = 174^2 \].

Step 4: Simplify the Equation

Combine like terms:\[ 2w^2 + 12w + 36 = 30276 \].

Step 5: Rearrange to Form a Quadratic Equation

Rearrange the terms to set the equation to zero:\[ 2w^2 + 12w + 36 - 30276 = 0 \].Simplify:\[ 2w^2 + 12w - 30240 = 0 \].

Step 6: Divide through by 2

Divide all terms by 2 to simplify:\[ w^2 + 6w - 15120 = 0 \].

Step 7: Solve the Quadratic Equation

Use the quadratic formula: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \], where \(a = 1\), \(b = 6\), and \(c = -15120\).Calculate the discriminant:\[ b^2 - 4ac = 6^2 - 4(1)(-15120) = 36 + 60480 = 60516 \].Find \( \sqrt{60516} \approx 246 \).

Step 8: Calculate the Two Possible Values for \(w\)

Substitute into the quadratic formula:\[ w = \frac{-6 \pm 246}{2} \].Calculate: \( w = \frac{-6 + 246}{2} = 120 \), and \( w = \frac{-6 - 246}{2} = -126 \).Since width can't be negative, \( w = 120 \) feet.

Step 9: Find the Length

The width \( w = 120 \) feet. Thus the length is \( w + 6 = 120 + 6 = 126 \) feet.

Key Concepts

Right Triangle

In geometry, a right triangle is a special type of triangle that has one angle measuring exactly 90 degrees. This right angle is the fundamental characteristic of right triangles.
The other two angles in a right triangle are always less than 90 degrees, and their measures add up to 90 degrees because the sum of all angles in any triangle is always 180 degrees.

• The side opposite the right angle is the longest side, known as the hypotenuse.
• The other two sides are called the legs.

Right triangles are crucial in many areas of math, including trigonometry and geometry. They serve as the basis for understanding more complex mathematical concepts.
Whenever you identify a right triangle, remember that you can apply the Pythagorean theorem, which provides a powerful tool for finding missing side lengths and solving many geometric problems.

Pythagorean Theorem

The Pythagorean theorem is an essential concept in mathematics that applies specifically to right triangles. Named after the ancient Greek mathematician Pythagoras, this theorem creates a relationship between the sides of a right triangle.
This theorem states:

• In a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

Mathematically, this is expressed as: \[a^2 + b^2 = c^2\]This equation allows you to solve for any one side of the triangle if you know the lengths of the other two sides.
If we consider the problem of finding the dimensions of a parcel using its diagonal, we see this theorem in action. The diagonal forms a hypotenuse of a right triangle, enabling us to use the Pythagorean theorem to set up an equation for finding the unknown sides.

Rectangular Diagonals

In rectangular geometry, diagonals are the straight lines that connect opposite corners of a rectangle. Understanding diagonals is critical, especially when dealing with the properties and measurements of rectangles and other shapes.

• Each rectangle has two equal diagonals.
• The diagonal divides the rectangle into two congruent right triangles.

By applying the concepts of right triangles and the Pythagorean theorem, the diagonal length can be used to derive the dimensions of the rectangle. In our problem, given the length of the diagonal, the width and length of the rectangle can be calculated.
Knowing how to handle and compute information about diagonals not only aids in solving geometry problems but also helps in practical applications in fields like architecture and construction, where precise measurements are crucial.