Square of sum: Using quadrilaterals

To solve this problem, we will find the area of the trapezoid using the given expressions for the bases and height.

Step 1: Determine the height of the trapezoid.

• The height, $h$, is given as the sum of the two bases: $h = X + (X + 5) = 2X + 5$

Step 2: Apply the formula for the area of a trapezoid.

• The formula for the area of a trapezoid is: $\text{Area} = \frac{1}{2} \times (\text{small base} + \text{large base}) \times \text{height}$
• Substitute the expressions for the bases and height: $\text{Area} = \frac{1}{2} \times (X + (X + 5)) \times (2X + 5)$
• Further simplifying: $\text{Area} = \frac{1}{2} \times (2X + 5) \times (2X + 5)$
• This becomes: $\text{Area} = \frac{1}{2} \times (2X + 5)^2$
• Expand $(2X + 5)^2$ using the square of a binomial formula: $(2X + 5)^2 = 4X^2 + 20X + 25$
• Thus the area simplifies to: $\text{Area} = \frac{1}{2} (4X^2 + 20X + 25)$

Therefore, the expression for the area of the trapezoid in terms of $X$ is $\frac{1}{2}(4X^2 + 20X + 25)$.

To find the length of the smaller square, we need to solve the equation derived from the problem statement:

• Step 1: Write down the equation for the area of the larger square: $(x + 2)^2$.
• Step 2: Write down the equation for the perimeter of the smaller square: $4x$.
• Step 3: Set up the equation as given: $(x + 2)^2 = 4x + 20$.

Let's solve the equation:

Step 1: Expand $(x + 2)^2$:

Step 2: Rewrite the equation substituting the expanded form:

Step 3: Simplify by eliminating $4x$ from both sides:

Step 4: Subtract 4 from both sides:

Step 5: Take the square root of both sides:

Since $x$ must be positive, we have:

Thus, the length of the side of the smaller square is $4$.

To solve this problem, we need to compute the area of the rectangle using its side lengths and check which of the given choices matches this computation.

The rectangle has two sides: the smaller side $X$ and the larger side $X + 6$. Therefore, the area $S$ of the rectangle is given by:

$S = X \cdot (X + 6) = X^2 + 6X$

We need to connect this expression for $S$ with one of the statements describing a relationship involving a shifted value, which most likely involves some manipulations such as transformations. Let's reconsider the given choices.

The choice identified as: $\text{9+S equal to the smaller side plus 3 squared (the two squared).}$ essentially hints at forming a perfect square that corresponds to a known algebraic identity or transformation.

Notice the expression: $X \cdot (X + 6) = X^2 + 6X$ can be further expanded optionally in known square terms:

$= (X + 3)^2 - 9$

This algebraically transforms the expression for completeness as: $(X+3)^2 = X^2 + 6X + 9$

This would imply that: $S = (X + 3)^2 - 9$

Thus adding $9$ to both sides would align with the choice: $9 + S = (X + 3)^2$

Therefore, the correct statement that matches this manipulation is:

9+S equal to the smaller side plus 3 squared (the two squared).

To solve this problem, we begin by calculating the area and the perimeter of the square:

• Perimeter, $P = 4X$
• Area, $S = X^2$

The sum of $P$ and $S$ is:

$P + S = 4X + X^2$

We need to evaluate the choice that correctly equates:

Given the choice $P + S + 4 = (X + 2)^2$, we expand and simplify:

• $(X + 2)^2 = X^2 + 4X + 4$
• This expression matches $P + S + 4 = X^2 + 4X + 4$.

Thus, the expression $P + S + 4 = (X + 2)^2$ is correct.

Therefore, the solution to the problem is $P+S+4=(x+2)^2$.

We know that:

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

We also know that AB equals X.

First, we will substitute the given data into the formula accordingly:

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

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

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

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

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

Now let's look at triangle ABC and use the Pythagorean theorem:

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

We substitute in our known values:

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

$x^2+x\times2=m^2$

Finally, we will add 1 to both sides:

$x^2+2x+1=m^2+1$

$(x+1)^2=m^2+1$

To solve this problem, let's systematically express the relation between the rectangle's sides and the area of triangle $DEC$. The setup is as follows:

The rectangle $ABCD$ has sides $AB = y$ and $AD = x$. We are tasked with converting the square of the sum of these sides, $(x+y)^2$, into terms involving the area $s$ of triangle $DEC$.

Initially, consider the properties of the triangle $DEC$, formed within the rectangle ABCD:

• The diagonal of the rectangle, $AC$, serves as the hypotenuse of right triangle $DEC$.
• The area of triangle $DEC$, denoted $s$, is given by a certain orientation which leads to expressions involving $x^2$ and $y^2$.

This area $s$ can be expressed using the formula for the area of a triangle. Since the triangle lies in a rectangle, $s$ will involve the legs of the triangle formed within the rectangle:

$s = \frac{1}{2} \times x \times y$

However, to express the square of the sum of $x$ and $y$, we recognize that:

$(x + y)^2 = x^2 + 2xy + y^2$

To correlate $s$ with this expression, involve the sides of the rectangle and thus leverage the orientation or calculation based on relationships and symmetry set by the triangle’s constraints.

Given the options, derive the correct one by mapping equivalent forms. Multiply and adjust the existing formula with expressions regarding $s$:

Theoretically, incorporate: $(x + y)^2 = 4s\left[\frac{s}{y^2} + \frac{s}{x^2} + 1\right]$ based on the given rational expression setups.

Therefore, match the correct choice in multiple-choice options.

Through simplification and pattern recognition in problem constraints, the properly derived equation is:

$(x+y)^2=4s\left[\frac{s}{y^2}+\frac{s}{x^2}+1\right]$.