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

Q: How do I find the area of triangle DEC from the rectangle?

Triangle DEC has vertices at D, E, and C. Since E is on side AB, you need to determine E's position first. The area formula is $ S = \frac{1}{2} \times base \times height $, where the base and height depend on E's coordinates.

Q: Why does the answer involve both X and Y in fractions?

The expression $ (X-Y)^2 $ expands to $ X^2 - 2XY + Y^2 $. When you substitute the relationships from the triangle area constraint, both X and Y appear in denominator terms because the area S relates to both rectangle dimensions.

Q: What does the constraint 'triangle area DEC equals S' tell us?

This constraint gives us a relationship between X, Y, and S. It means we can express one variable in terms of the others, like $ XY = 2S $, which we then substitute into our target expression.

Q: How do I expand and simplify the final expression?

Start with $ (X-Y)^2 = X^2 - 2XY + Y^2 $. Use the constraint to substitute XY = 2S, then express X and Y individually using the area relationship to get the final form with S terms.

Q: Can I verify my answer is correct?

Yes! Substitute specific values that satisfy your constraints (like X=4, Y=2, S=4) into both the original expression $ (X-Y)^2 $ and your final answer. Both should give the same numerical result.

Algebraic Expressions with Triangle Area Constraints

Given the rectangle ABCD

AB=Y AD=X

The triangular area DEC equals S:

Express the square of the difference of the sides of the rectangle

using X, Y and S:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the square of the rectangle side differences

00:11 Substitute in the side expressions according to the given data and proceed to open parentheses

00:20 Triangle area expression according to the given data

00:27 Triangle height equals rectangle side X

00:30 Apply the formula for calculating the area of the triangle

00:33 (height x base) divided by 2

00:40 Determine the expression for X

00:46 Determine the expression for Y

00:54 Determine the expression for the product of the sides

01:02 Substitute these expressions into our shortened multiplication formula

01:17 Pay attention to squaring both the numerator and the denominator

01:29 Remove the common factor from the parentheses

01:44 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=Y AD=X

The triangular area DEC equals S:

Express the square of the difference of the sides of the rectangle

using X, Y and S:

Step-by-step solution

Since we are given the length and width, we will substitute them according to the formula:

$(AD-AB)^2=(x-y)^2$

$x^2-2xy+y^2$

The height is equal to side AD, meaning both are equal to X

Let's calculate the area of triangle DEC:

$S=\frac{xy}{2}$

$x=\frac{2S}{y}$

$y=\frac{2S}{x}$

$xy=2S$

Let's substitute the given data into the formula above:

$(\frac{2S}{y})^2-2\times2S+(\frac{25}{x})^2$

$\frac{4S^2}{y^2}-4S+\frac{4S^2}{x^2}$

$4S=(\frac{S^2}{y}+\frac{S^2}{x}-1)$

Final Answer

$(x-y)^2=4s\lbrack\frac{s}{y^2}+\frac{s}{x^2}-1\rbrack$

Key Points to Remember

Essential concepts to master this topic

Rectangle Properties: Use side lengths X and Y to establish coordinate relationships
Triangle Area Formula: Calculate $S = \frac{1}{2} \times base \times height$ for triangle DEC
Verification Method: Substitute back into original expression to confirm $(X-Y)^2 = 4S[\frac{S}{Y^2}+\frac{S}{X^2}-1]$ ✓

Common Mistakes

Avoid these frequent errors

Incorrectly calculating triangle DEC area
Don't assume triangle DEC has area $\frac{XY}{2}$ = wrong setup! This ignores point E's position on side AB. Always identify the correct base and height: E divides AB, so the triangle's base and height depend on E's specific location.

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 find the area of triangle DEC from the rectangle?

Triangle DEC has vertices at D, E, and C. Since E is on side AB, you need to determine E's position first. The area formula is $S = \frac{1}{2} \times base \times height$ , where the base and height depend on E's coordinates.

Why does the answer involve both X and Y in fractions?

The expression $(X-Y)^2$ expands to $X^2 - 2XY + Y^2$ . When you substitute the relationships from the triangle area constraint, both X and Y appear in denominator terms because the area S relates to both rectangle dimensions.

What does the constraint 'triangle area DEC equals S' tell us?

This constraint gives us a relationship between X, Y, and S. It means we can express one variable in terms of the others, like $XY = 2S$ , which we then substitute into our target expression.

How do I expand and simplify the final expression?

Start with $(X-Y)^2 = X^2 - 2XY + Y^2$ . Use the constraint to substitute XY = 2S, then express X and Y individually using the area relationship to get the final form with S terms.

Can I verify my answer is correct?

Yes! Substitute specific values that satisfy your constraints (like X=4, Y=2, S=4) into both the original expression $(X-Y)^2$ and your final answer. Both should give the same numerical result.

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

Algebraic Expressions with Triangle Area Constraints

❤️ 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 find the area of triangle DEC from the rectangle?

Why does the answer involve both X and Y in fractions?

What does the constraint 'triangle area DEC equals S' tell us?

How do I expand and simplify the final expression?

Can I verify my answer is correct?

🌟 Unlock Your Math Potential

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