Question

Find the length of the diagonal of a rectangle whose length is 8 meters and whose width is 5 meters.

Short answer

The diagonal of the rectangle is approximately 9.43 meters.

Step-by-step solution

Identify the Formula

To find the diagonal of a rectangle, you can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides (length and width). The formula is \( d = \sqrt{l^2 + w^2} \), where \( d \) is the diagonal, \( l \) is the length, and \( w \) is the width.

Plug in the Values

Substitute the given values for the length and width into the formula. Here, the length (\( l \)) is 8 meters and the width (\( w \)) is 5 meters, so the equation becomes \( d = \sqrt{8^2 + 5^2} \).

Simplify the Equation

Calculate the squares of the length and width. This gives you \( d = \sqrt{64 + 25} \).

Add the Values

Add the squares of the length and width to get \( d = \sqrt{89} \).

Calculate the Square Root

Compute the square root of 89 to find the length of the diagonal. Thus, \( d = \sqrt{89} \approx 9.43 \) meters.

Key concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that deals with right-angled triangles. This theorem is named after the ancient Greek mathematician Pythagoras. It states that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides, commonly referred to as the legs of the triangle.

Mathematically, it is expressed as:\[c^2 = a^2 + b^2\]where:

• \(c\) is the length of the hypotenuse.
• \(a\) and \(b\) are the lengths of the other two sides.

In the context of this exercise, when we need to calculate the diagonal of a rectangle, we can treat the length and width of the rectangle as the two shorter sides of a right triangle. This allows us to use the Pythagorean theorem to find the length of the diagonal which acts as the hypotenuse.

Knowing how to apply this theorem can be incredibly useful in a variety of geometric problems, especially those involving any shape that can be divided into right-angled triangles.

Rectangle properties

A rectangle is a quadrilateral with four right angles. It is a specialized form of a parallelogram and exhibits some unique properties which make it particularly straightforward to solve problems associated with it.

Here are some of the essential properties of rectangles:

• Opposite sides are equal in length.
• The diagonals of a rectangle are equal in length.
• Each angle measures 90 degrees.

For this exercise, the most relevant property is that the diagonals are equal. Because the diagonals bisect each other, the rectangle can be perceived as two congruent right-angled triangles. This makes using the Pythagorean theorem possible for calculating the diagonal.

Understanding these properties helps provide confidence when solving problems as you know the relationships between the sides and angles of the rectangle.

Diagonal calculation

Calculating the diagonal of a rectangle involves using both its length and width in combination with the Pythagorean theorem. In geometry, the diagonal of a rectangle refers to the segment connecting opposite corners, creating two equal triangles. Each diagonal offers a clear visual representation of how a rectangle can be divided, demonstrating the close relationship between its length, width, and diagonal.

To calculate this diagonal using the rectangle with length (\(l\)) of 8 meters and width (\(w\)) of 5 meters, apply the formula:\[d = \sqrt{l^2 + w^2}\]Step-by-step, this solution involves:

• Plugging in the rectangle's length and width: \(d = \sqrt{8^2 + 5^2}\)
• Calculating the squares: \(8^2 = 64\) and \(5^2 = 25\)
• Adding the squares to get \(64 + 25 = 89\)
• Finding the square root of 89, which is approximately 9.43 meters.

This method is an excellent demonstration of how geometry uses basic principles to solve practical problems efficiently.