Area of a Rectangle: Using short multiplication formulas

First, recall the formula for calculating the area of a rectangle:

The area of a rectangle (which has two pairs of equal opposite sides and all angles are $90\degree$) with sides of length $a,\hspace{4pt} b$ units, is given by the formula:

$\boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b }$(square units)

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After recalling the formula for the area of a rectangle, let's proceed to solve the problem:

Begin by denoting the area of the given rectangle as: $S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}$ and proceed to write (in mathematical notation) the given information:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-13$

Continue to calculate the area of the rectangle given in the problem:

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Using the rectangle area formula mentioned earlier:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x-4)(x+1)$

Continue to simplify the expression that we obtained for the rectangle's area, using the distributive property:

$(c+d)(h+g)=ch+cg+dh+dg$

We are able to obtain the area of the rectangle by

using the distributive property as shown below:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x-4)(x+1) \\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2+x-4x-4\\ \boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-3x-4}$

Recall the given information:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-13$

Therefore, we can conclude that:

$x^2-3x-4=x^2-13 \\ \downarrow\\ -3x=4-13\\ -3x=-9\hspace{6pt}\text{/}(-3)\\ \boxed{x=3}$

We solved the resulting equation simply by combining like terms, isolating the expression with the unknown on one side and dividing both sides by the unknown's coefficient in the final step,

Note that this result satisfies the domain of definition for x, which was given as:

-1\text{<}x\text{<}4 and therefore it is the correct result

The correct answer is answer C.

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:

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

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

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:

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

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

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:

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

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:

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

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

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