Rectangle ABCD: Finding Diagonal Length m with Side Ratio √(x/2)

Q: How do I handle the square root in the ratio?

Start by writing the ratio as $ \frac{x}{BC} = \frac{\sqrt{x}}{\sqrt{2}} $. Then cross-multiply to get $ \sqrt{2} \cdot x = \sqrt{x} \cdot BC $, and solve for BC.

Q: Why do we divide by square root x in the solution?

We divide both sides by $ \sqrt{x} $ to isolate BC. Remember: $ \frac{\sqrt{2} \cdot x}{\sqrt{x}} = \sqrt{2} \cdot \sqrt{x} $ because x can be written as $ \sqrt{x} \cdot \sqrt{x} $.

Q: How does the Pythagorean theorem apply here?

Since ABCD is a rectangle, triangle ABC is a right triangle. The diagonal AC is the hypotenuse, so we use: $ AB^2 + BC^2 = AC^2 $ or $ x^2 + (\sqrt{2x})^2 = m^2 $.

Q: What does $ (\sqrt{2x})^2 $ equal?

When you square a square root, they cancel out! So $ (\sqrt{2x})^2 = 2x $. This is why our final equation becomes $ x^2 + 2x = m^2 $.

Q: Can I check my answer using specific values?

Yes! Try x = 4: then $ BC = \sqrt{2 \cdot 4} = \sqrt{8} = 2\sqrt{2} $. Check the ratio: $ \frac{4}{2\sqrt{2}} = \sqrt{2} = \sqrt{\frac{4}{2}} $ ✓

Pythagorean Theorem with Square Root Ratios

Given the rectangle ABCD

AB=X the ratio between AB and BC is equal to $\sqrt{\frac{x}{2}}$

We mark the length of the diagonal $A$ with $m$

Check the correct argument:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Mark the correct statement

00:03 Side ratio according to the given data

00:11 Insert appropriate values into the expression and solve to find BC

00:17 Make sure to extract the root for both the numerator and denominator

00:30 Isolate BC

00:38 Factor X into the square root of X multiplied by the square root of X

00:44 Simplify wherever possible

00:49 This is the length of side BC

01:03 Use the Pythagorean theorem in triangle ABC

01:10 Insert the appropriate values into the expression and solve

01:25 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the rectangle ABCD

AB=X the ratio between AB and BC is equal to $\sqrt{\frac{x}{2}}$

We mark the length of the diagonal $A$ with $m$

Check the correct argument:

Step-by-step solution

Let's find side BC

Based on what we're given:

$\frac{AB}{BC}=\frac{x}{BC}=\sqrt{\frac{x}{2}}$

$\frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}$

$\sqrt{2}x=\sqrt{x}BC$

Let's divide by square root x:

$\frac{\sqrt{2}\times x}{\sqrt{x}}=BC$

$\frac{\sqrt{2}\times\sqrt{x}\times\sqrt{x}}{\sqrt{x}}=BC$

Let's reduce the numerator and denominator by square root x:

$\sqrt{2}\sqrt{x}=BC$

We'll use the Pythagorean theorem to calculate the area of triangle ABC:

$AB^2+BC^2=AC^2$

Let's substitute what we're given:

$x^2+(\sqrt{2}\sqrt{x})^2=m^2$

$x^2+2x=m^2$

Final Answer

$x^2+2x=m^2$

Key Points to Remember

Essential concepts to master this topic

Ratio Analysis: Convert given ratio to find the unknown side length
Technique: From $\frac{x}{BC} = \sqrt{\frac{x}{2}}$ , solve to get $BC = \sqrt{2x}$
Check: Verify using Pythagorean theorem: $x^2 + 2x = m^2$ ✓

Common Mistakes

Avoid these frequent errors

Incorrectly simplifying the ratio equation
Don't just cross-multiply $\frac{x}{BC} = \sqrt{\frac{x}{2}}$ without careful algebra = wrong side length! Students often make errors when manipulating square roots. Always isolate BC step-by-step: multiply both sides by BC, then divide by the square root term.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side DC has a length of 1.5 cm and side AD has a length of 9.5 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

How do I handle the square root in the ratio?

Start by writing the ratio as $\frac{x}{BC} = \frac{\sqrt{x}}{\sqrt{2}}$ . Then cross-multiply to get $\sqrt{2} \cdot x = \sqrt{x} \cdot BC$ , and solve for BC.

Why do we divide by square root x in the solution?

We divide both sides by $\sqrt{x}$ to isolate BC. Remember: $\frac{\sqrt{2} \cdot x}{\sqrt{x}} = \sqrt{2} \cdot \sqrt{x}$ because x can be written as $\sqrt{x} \cdot \sqrt{x}$ .

How does the Pythagorean theorem apply here?

Since ABCD is a rectangle, triangle ABC is a right triangle. The diagonal AC is the hypotenuse, so we use: $AB^2 + BC^2 = AC^2$ or $x^2 + (\sqrt{2x})^2 = m^2$ .

What does $(\sqrt{2x})^2$ equal?

When you square a square root, they cancel out! So $(\sqrt{2x})^2 = 2x$ . This is why our final equation becomes $x^2 + 2x = m^2$ .

Can I check my answer using specific values?

Yes! Try x = 4: then $BC = \sqrt{2 \cdot 4} = \sqrt{8} = 2\sqrt{2}$ . Check the ratio: $\frac{4}{2\sqrt{2}} = \sqrt{2} = \sqrt{\frac{4}{2}}$ ✓

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Rectangle ABCD: Finding Diagonal Length m with Side Ratio √(x/2)

Pythagorean Theorem with Square Root Ratios

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I handle the square root in the ratio?

Why do we divide by square root x in the solution?

How does the Pythagorean theorem apply here?

What does $(\sqrt{2x})^2$ equal?

Can I check my answer using specific values?

🌟 Unlock Your Math Potential

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❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I handle the square root in the ratio?

Why do we divide by square root x in the solution?

How does the Pythagorean theorem apply here?

What does (2x)2 (\sqrt{2x})^2 (2x​)2 equal?

Can I check my answer using specific values?

🌟 Unlock Your Math Potential

More Questions

Ratio

Calculate 2/3 of 18: Finding White Balls in a Box

Calculate 1/4 of 28: Finding Orange Balls in a Box

Triangle Area 10 cm²: Finding Base X When Height is 5 Times Greater

Triangle Area with 5/8 Ratio: Calculate Using 16-Unit Height

Area of a Rectangle

Rectangle Area Problem: Finding Side Length When Area = 55 and Height = 5.5

Rectangle Area Problem: Finding Width When Area = 15 and Height = 2.5

Calculate Circle Area When Rectangle Area = 28X: Geometric Relationship Problem

Calculate Rectangle Side Length: Finding AB When Area = 27 and Height = 3

Square of Difference

Solve Complex Rational Equation: ((2x-1)²)/(x-2) + ((x-2)²)/(2x-1) = 4.5x

Solve the Equation: 1/(x-2)² + 1/(x-2) = 1 Step-by-Step

Rectangle ABCD: Express (X-Y)² Using Side Lengths and Triangle Area S

Trapezoid Area Formula: Height Equals Sum of Bases with X-5 Difference

Product Property of Square Roots

Multiply Square Roots: Finding √a × √b Expression

Solve sqrt(3) × sqrt(12) + 3²: Complete Radical Expression Calculation

Diagonals

Rectangle Side Length: Using Pythagorean Theorem with AC=13 and AB=12

Calculate the Perimeter of Rectangle ABCD with Sides 24 and 25 Units

What does $(\sqrt{2x})^2$ equal?