Problem 3
Question
For Problems 1 through 7, give exact answers, not numerical approximations. How long is the diagonal of a square whose sides are 5 inches?
Step-by-Step Solution
Verified Answer
The length of the diagonal of a square is \( 5 \sqrt{2} \) inches.
1Step 1: Understand the Problem
We need to calculate the length of the diagonal (d) of a square with 5-inch sides. As stated, all sides of a square are equal, so the two sides that form the right angle will each be 5 inches. We can use the Pythagorean theorem, which in this case will be \( a^2 + b^2 = c^2 \). Since a and b are both 5 inches, we will substitute 5 in for a and b. The equation then becomes \( 5^2 + 5^2 = c^2 \).
2Step 2: Applying the Pythagorean theorem
Applying the Pythagorean theorem, we calculate (5 units)^2 + (5 units)^2 to get \( c^2 \), which equals 25 + 25 = 50. Therefore, \( c^2 = 50 \).
3Step 3: Find Length of the Diagonal
\( c^2 = 50 \) from the previous equation. To solve for c, we take the square root of both sides to find that c equals the square root of 50.
4Step 4: Simplifying the Radical Expression
The square root of 50 equals \( 5 \sqrt{2} \).
Key Concepts
Pythagorean TheoremSquare RootRight Angle Triangle
Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry, particularly useful when dealing with right angle triangles. It states that in a right 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. This can be expressed in the equation:
\[\begin{equation}a^2 + b^2 = c^2\text{,}\end{equation}\]
where 'a' and 'b' are the lengths of the two legs of the triangle, and 'c' represents the hypotenuse. When calculating the diagonal of a square, this theorem is particularly handy as the diagonal itself acts as the hypotenuse, slicing the square into two right angle triangles. Therefore, by knowing the lengths of the square’s sides, one can easily calculate the diagonal's length by rearranging and solving the Pythagorean theorem.
\[\begin{equation}a^2 + b^2 = c^2\text{,}\end{equation}\]
where 'a' and 'b' are the lengths of the two legs of the triangle, and 'c' represents the hypotenuse. When calculating the diagonal of a square, this theorem is particularly handy as the diagonal itself acts as the hypotenuse, slicing the square into two right angle triangles. Therefore, by knowing the lengths of the square’s sides, one can easily calculate the diagonal's length by rearranging and solving the Pythagorean theorem.
Square Root
The square root is a mathematical operation that, simply put, asks the question 'What number, multiplied by itself, gives me this value?'. When we take the square root of a number 'x', we are looking for another number, which when squared, returns 'x'. This is symbolically represented as
\[\begin{equation} \sqrt{x} \text{.}\end{equation}\]
It's the inverse operation of squaring a number. In the context of our problem with the square's diagonal, once we have used the Pythagorean theorem to obtain the square of the diagonal (hypotenuse), we then need to apply the square root function to find its actual length. For example, having obtained the value of the diagonal squared as 50 (\(c^2 = 50\)), we are interested in the value of 'c', which would be \(\sqrt{50}\), or simplifying further, \(5\sqrt{2}\), as we are looking for the number that, when squared, equals 50.
\[\begin{equation} \sqrt{x} \text{.}\end{equation}\]
It's the inverse operation of squaring a number. In the context of our problem with the square's diagonal, once we have used the Pythagorean theorem to obtain the square of the diagonal (hypotenuse), we then need to apply the square root function to find its actual length. For example, having obtained the value of the diagonal squared as 50 (\(c^2 = 50\)), we are interested in the value of 'c', which would be \(\sqrt{50}\), or simplifying further, \(5\sqrt{2}\), as we are looking for the number that, when squared, equals 50.
Right Angle Triangle
A right angle triangle is a type of triangle characterized by having one angle that measures exactly 90 degrees. The two sides forming this angle are referred to as the 'legs' of the triangle, while the side opposite the right angle is called the 'hypotenuse'. The special property of a right angle triangle allows for the application of the Pythagorean theorem to find the relationship between its sides.
Identifying a Right Angle Triangle
The right angle can be recognized by a small square placed at the vertex of the angle. This signifies a 90-degree angle, distinguishing it from other types of triangles. In problems like finding the diagonal of a square, each half of the square created by drawing the diagonal forms two identical right angle triangles, which share the diagonal as the common hypotenuse. By using the properties of right angles and the theorem, we can accurately determine measurements within geometric shapes—perfect for tackling geometry homework or understanding spatial relationships.Other exercises in this chapter
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