Find the side length of the square with diagonal 10 cm. Write as a radical.
Understand the Problem
The question is asking for the calculation of the side length of a square given its diagonal, which is 10 cm. The task is to express the side length as a radical.
Answer
$5\sqrt{2}$ cm
Answer for screen readers
The side length of the square is $5\sqrt{2}$ cm.
Steps to Solve
- Understanding the Relationship of Side and Diagonal
In a square, the diagonal $d$ can be calculated using the formula:
$$ d = a\sqrt{2} $$
where $a$ is the length of a side of the square.
- Setting Up the Equation
Given that the diagonal $d$ is 10 cm, we substitute this value into the equation:
$$ 10 = a\sqrt{2} $$
- Solving for Side Length
To find the side length $a$, we can rearrange the equation:
$$ a = \frac{10}{\sqrt{2}} $$
- Rationalizing the Denominator
Rationalizing the denominator involves multiplying the numerator and denominator by $\sqrt{2}$:
$$ a = \frac{10 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{10\sqrt{2}}{2} $$
- Simplifying the Expression
Now simplify the fraction:
$$ a = 5\sqrt{2} $$
The side length of the square is $5\sqrt{2}$ cm.
More Information
The relationship between the side length and the diagonal of a square is crucial in geometry. The formula derives from the Pythagorean theorem, as each diagonal splits the square into two right triangles.
Tips
- Forgetting to rationalize the denominator can lead to an unsimplified answer.
- Mixing up the formula; some may mistakenly use $d = 2a$ for the diagonal.
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