Problem 85 - Algebra and Trigonometry

Question

Dimensions of a Lot A city lot has the shape of a right triangle whose hypotenuse is 7 ft longer than one of the other sides. The perimeter of the lot is 392 ft. How long is each side of the lot?

Step-by-Step Solution

Answer

The sides are approximately 119 ft, 120 ft, and 153 ft.

Step 1: Identify the Known Variables

Let the two perpendicular sides of the right triangle be \(a\) and \(b\), and the hypotenuse be \(c\). We know from the problem statement that \(c = b + 7\). The perimeter is given as 392 ft, so we have the equation \(a + b + c = 392\).

Step 2: Express All Sides in Terms of One Variable

Substitute \(c = b + 7\) into the perimeter equation: \(a + b + (b + 7) = 392\), which simplifies to \(a + 2b + 7 = 392\). Solve for \(a\) in terms of \(b\): \(a = 385 - 2b\).

Step 3: Use the Pythagorean Theorem Relation

In a right triangle, the sides satisfy \(a^2 + b^2 = c^2\). Substitute \(a = 385 - 2b\) and \(c = b + 7\) into this equation: \((385 - 2b)^2 + b^2 = (b + 7)^2\).

Step 4: Simplify and Solve the Quadratic Equation

Expand the squares: \((385 - 2b)^2 = 148225 - 1540b + 4b^2\)\((b + 7)^2 = b^2 + 14b + 49\).Set up the equation: \(148225 - 1540b + 4b^2 + b^2 = b^2 + 14b + 49\).Simplify to find a quadratic equation for \(b\): \(4b^2 - 1540b + 148225 = 14b + 49\).This simplifies to: \(4b^2 - 1554b + 148176 = 0\).

Step 5: Solve the Quadratic Equation

Use the quadratic formula \(b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\) with \(A = 4, B = -1554, C = 148176\):\(b = \frac{-(-1554) \pm \sqrt{1554^2 - 4 \cdot 4 \cdot 148176}}{2 \cdot 4}\).Calculate the determinant and solve for \(b\). Find the realistic positive root for \(b\).

Step 6: Calculate the Length of Side a and Hypotenuse c

Once \(b\) is found, calculate \(a = 385 - 2b\) and \(c = b + 7\). Verify the solution by checking that these values satisfy the Pythagorean relationship and the perimeter condition.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the sides of a right triangle.
It's expressed as:

• \(a^2 + b^2 = c^2\)

where \(a\), \(b\), and \(c\) are the sides of the triangle, with \(c\) as the hypotenuse, the longest side opposite the right angle.
This relationship helps in finding unknown side lengths when the other two are known. In the problem involving the city lot, the theorem aids in formulating an equation using the given characteristics of the triangle.
For instance, by substituting expressions obtained from perimeter considerations into \(a^2 + b^2 = c^2\), the unknowns become manageable within a solvable quadratic equation.
This demonstrates the versatility and importance of the Pythagorean Theorem in solving real-world geometric problems.

Quadratic Equation

Quadratic equations appear commonly in mathematical problems involving squared terms, especially when using the Pythagorean Theorem in right triangles.
The standard form is:

• \(Ax^2 + Bx + C = 0\)

where \(A\), \(B\), and \(C\) are coefficients, and \(x\) represents the unknown variable. In solving the right triangle problem, once expressions for sides \(a\), \(b\), and \(c\) in terms of a single variable are established,
a quadratic equation emerges.
To solve it, the quadratic formula is applied:

• \(x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\)

The discriminant \(B^2 - 4AC\) decides the nature of the roots:

• If positive, there are two distinct real roots.
• If zero, there is one real root.
• If negative, there are no real roots.

In our problem, solving the quadratic yields the length of side \(b\), which is crucial for determining the other two sides.

Perimeter

Perimeter is the total length around a 2-dimensional shape. In triangles, it's simply the sum of all three sides:

• \(P = a + b + c\)

For the right triangle problem, the perimeter value of 392 ft provides a pivotal equation to relate the sides:

• \(a + b + c = 392\)

By substituting \(c\) with \(b+7\) as given, we further simplify to find \(a\) and \(b\) in terms of each other.Understanding and setting up correct perimeter equations are crucial for integrating them with other given conditions. This step facilitates breaking down the problem into solvable parts, such as utilizing the Pythagorean relationship to form and solve a quadratic equation. Solving these perimeter equations correctly ensures that all conditions in the problem are met precisely.