Problem 95 - Elementary Algebra

Question

The length of a rectangle is twice its width. If the diagonal measures 10 feet, then find the dimensions of the rectangle.

Step-by-Step Solution

Answer

The rectangle's width is \( 2\sqrt{5} \) feet and its length is \( 4\sqrt{5} \) feet.

Step 1: Define Variables

Let the width of the rectangle be represented by \( w \). Since the length is twice the width, we can express the length as \( 2w \).

Step 2: Apply the Pythagorean Theorem

A rectangle's diagonal, length, and width form a right triangle. According to the Pythagorean theorem, the diagonal squared equals the sum of the squares of the width and length. Therefore, we have the equation \( w^2 + (2w)^2 = 10^2 \).

Step 3: Simplify the Equation

Simplify \( w^2 + (2w)^2 = 10^2 \) to get \( w^2 + 4w^2 = 100 \), which simplifies further to \( 5w^2 = 100 \).

Step 4: Solve for Width

Divide both sides of the equation by 5 to solve for \( w^2 \), resulting in \( w^2 = 20 \). Taking the square root of both sides gives \( w = \sqrt{20} = 2\sqrt{5} \).

Step 5: Find the Length

Since the length is twice the width, substitute \( w = 2\sqrt{5} \) into \( 2w \) to find the length: \( 2(2\sqrt{5}) = 4\sqrt{5} \).

Step 6: Conclude the Dimensions

Thus, the dimensions of the rectangle are \( 2\sqrt{5} \) feet for the width and \( 4\sqrt{5} \) feet for the length.

Key Concepts

Rectangles

Rectangles are one of the fundamental shapes in geometry. They are four-sided polygons with opposite sides that are equal, and all angles are right angles, meaning they measure 90 degrees.
When dealing with rectangles, it is important to remember these basic properties, as they can simplify various problems.
One particular property of a rectangle is that its diagonals are equal in length. This means that the line segment connecting opposite corners is the same for both sets of corners.

• The diagonal of a rectangle can be calculated using the Pythagorean Theorem if the length and the width are known.
• These properties make calculating areas and perimeters straightforward, as the area is simply the width times the length, and the perimeter is twice the length plus twice the width.

By understanding these properties, solving problems related to rectangles becomes a much simpler task.

Algebraic Equations

Algebraic equations are mathematical statements showing the equality between two expressions, often involving a variable to solve for.
In geometry and problems involving shapes like rectangles, algebraic equations can often express relationships between dimensions.
For instance, in our example problem, the relationship between width and length of a rectangle can be defined using algebraic terms: the length is twice the width, expressed as \( 2w \).

• Equations are used to model real-world situations, allowing one to solve problems by finding unknown values.
• In many cases, especially with shapes or motion, equations derived from the Pythagorean Theorem are employed to find missing dimensions, as illustrated in the exercise above.

It’s crucial to effectively manipulate and solve these equations, as this skill aids in uncovering the information logic behind a problem's layout.

Geometry

Geometry involves studying shapes, sizes, and the properties of space. One of the core elements of geometry is dealing with different types of figures like rectangles, triangles, and circles.
Geometric principles come in handy when solving problems related to shapes.
For example, when solving for the dimensions of a rectangle based on its diagonal, the Pythagorean Theorem does a lot of heavy lifting.

• The Pythagorean Theorem is key in these situations, relating the sides of a right triangle with its hypotenuse: \( a^2 + b^2 = c^2 \).
• This theorem applies to rectangles because any rectangle with a diagonal can be divided into two right triangles.

With these foundational ideas in place, one can approach problems with greater understanding and efficiency, drawing logical conclusions based on geometric relationships.