Square Diagonal Formula: Finding Length in Terms of Side 'a'

Q: Why can't the diagonal just be 2a?

The diagonal is not twice the side! It's the hypotenuse of a right triangle. Think of it this way: if you walk along two sides of a square, you travel distance 2a, but cutting diagonally is a shorter path.

Q: Where does the √2 come from?

When you apply the Pythagorean theorem: $ c^2 = a^2 + a^2 = 2a^2 $. Taking the square root of both sides gives $ c = \sqrt{2a^2} = \sqrt{2} \cdot a $.

Q: Is √2a the same as 2√a?

No! $ \sqrt{2}a $ means $ \sqrt{2} \times a $, while $ 2\sqrt{a} $ means $ 2 \times \sqrt{a} $. These are completely different expressions!

Q: Can I use this formula for rectangles too?

Only for squares! For rectangles with different side lengths, you need $ d = \sqrt{l^2 + w^2} $ where l and w are the different side lengths.

Q: What's the approximate value of √2?

$ \sqrt{2} \approx 1.414 $. So if your square has side length 5, the diagonal is approximately 5 × 1.414 = 7.07 units.

Pythagorean Theorem with Square Diagonals

A square has sides measuring a.

Choose the function that expresses the length of the square's diagonal.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the diagonal of the square

00:03 In a square all sides are equal

00:08 Let's use the Pythagorean theorem on the triangle

00:13 We'll substitute appropriate values according to the given data, and calculate to find the diagonal

00:21 We'll extract the root

00:28 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A square has sides measuring a.

Choose the function that expresses the length of the square's diagonal.

Step-by-step solution

To find the length of the diagonal of a square with side length $a$ , we will use the Pythagorean theorem.

A diagonal divides a square into two congruent right-angled triangles. The legs of these triangles are the sides of the square, both with length $a$ . The diagonal then serves as the hypotenuse.

According to the Pythagorean theorem, the hypotenuse $c$ of a right triangle with legs $a$ is found via:

$c = \sqrt{a^2 + a^2}$

Simplifying inside the square root gives us:

$c = \sqrt{2a^2}$

We can further simplify this expression:

$c = \sqrt{2} \cdot \sqrt{a^2} = \sqrt{2}a$

Thus, the length of the diagonal is expressed by the function $y = \sqrt{2}a$ .

In this problem, this solution corresponds to choice 1, which is $y = \sqrt{2}a$ .

The chosen answer is correct as verified by the application of the Pythagorean theorem to a square.

Therefore, the correct function for the diagonal is $y = \sqrt{2}a$ .

Final Answer

$y=\sqrt{2}a$

Key Points to Remember

Essential concepts to master this topic

Rule: Square diagonal creates two right triangles with equal legs
Technique: Apply $c = \sqrt{a^2 + a^2} = \sqrt{2a^2} = \sqrt{2}a$
Check: Verify diagonal is approximately 1.414 times the side length ✓

Common Mistakes

Avoid these frequent errors

Confusing diagonal with side or perimeter
Don't just double the side length (2a) = wrong answer! The diagonal is longer than any side but shorter than the perimeter. Always use the Pythagorean theorem to find the hypotenuse of the right triangle formed.

Practice Quiz

Test your knowledge with interactive questions

Complete:

The missing value of the function point:

$$ f(x)=x^2 $$

$$ f(?)=16 $$

FAQ

Everything you need to know about this question

Why can't the diagonal just be 2a?

The diagonal is not twice the side! It's the hypotenuse of a right triangle. Think of it this way: if you walk along two sides of a square, you travel distance 2a, but cutting diagonally is a shorter path.

Where does the √2 come from?

When you apply the Pythagorean theorem: $c^2 = a^2 + a^2 = 2a^2$ . Taking the square root of both sides gives $c = \sqrt{2a^2} = \sqrt{2} \cdot a$ .

Is √2a the same as 2√a?

No! $\sqrt{2}a$ means $\sqrt{2} \times a$ , while $2\sqrt{a}$ means $2 \times \sqrt{a}$ . These are completely different expressions!

Can I use this formula for rectangles too?

Only for squares! For rectangles with different side lengths, you need $d = \sqrt{l^2 + w^2}$ where l and w are the different side lengths.

What's the approximate value of √2?

$\sqrt{2} \approx 1.414$ . So if your square has side length 5, the diagonal is approximately 5 × 1.414 = 7.07 units.

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