The diagonal of a square is $x$ units. What is the area of the square in terms of $x$?
A. $\frac{1}{2} x^2$ square units
B. $x^2$ square units
C. $2 x$ square units
D. $\frac{1}{2} x$ square units
Answer (2)
Define the side length of the square as s and use the Pythagorean theorem to relate it to the diagonal x : s 2 + s 2 = x 2 .
Simplify the equation to find 2 s 2 = x 2 .
Solve for s 2 in terms of x : s 2 = 2 x 2 .
Express the area of the square as A = s 2 = 2 1 x 2 .
Explanation
Define variables and state the goal. Let s be the side length of the square. We are given that the diagonal of the square is x units. We need to find the area of the square in terms of x .
Apply the Pythagorean theorem. Using the Pythagorean theorem, we can relate the side length s and the diagonal x of the square. The Pythagorean theorem states that in a 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. In our case, the diagonal of the square is the hypotenuse of a right-angled triangle formed by two sides of the square. Therefore, we have: s 2 + s 2 = x 2
Simplify the equation. Simplifying the equation, we get: 2 s 2 = x 2
Solve for s^2. Now, we solve for s 2 in terms of x :
s 2 = 2 x 2
Find the area of the square. The area of the square is given by A = s 2 . Since we found that s 2 = 2 x 2 , the area of the square in terms of x is: A = 2 1 x 2 Therefore, the area of the square is 2 1 x 2 square units.
Examples
Understanding the relationship between a square's diagonal and its area is useful in various fields, such as construction and design. For example, if you're designing a square patio and know the length of the diagonal you want, you can easily calculate the area of paving stones needed using the formula A = 2 1 x 2 , where x is the diagonal length. This helps in estimating material costs and planning the layout efficiently.
The area of a square, given its diagonal x , can be found using the formula A = 2 1 x 2 square units. This is derived using the Pythagorean theorem and relates the side length and the diagonal of the square. Therefore, the answer is option A: 2 1 x 2 .
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