Q45. - Algebra 2

Question

What is the area of square $ABCD$.

$\begin{array}{l}\left(A\right)25{\text{ unit}}^2\\ \left(B\right)4\sqrt{29}{\text{ unit}}^2\\ \left(C\right)29{\text{ unit}}^2\\ \left(D\right)25+\sqrt{2}{\text{ unit}}^2\end{array}$

Step-by-Step Solution

Answer

The correct option is C, because the area of the square $ABCD$ is $29{\text{ unit}}^2$.

Step 1 – The distance formula.

The distance between two points $\left(x_1,y_1\right),$ and $\left(x_2,y_2\right)$ is $D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$.

Step 2 – Find any two edge points of any side of the given square.

Here, $ABCD$ is a square, so, the lengths of the four sides $AB,BC,CD,$ and $DA$ are equal.

From the graph, the endpoints of the side $D and A$ are $\left(0,-5\right)$ and $\left(-2,0\right)$.

Step 3 – Find the length of the side D A .

Here, $\left(x_1,y_1\right)=\left(0,-5\right)$ and $\left(x_2,y_2\right)=\left(-2,0\right)$.

Find the length $DA:$

 $\begin{array}{c}D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(-2-0\right)}^2+{\left(0-\left(-5\right)\right)}^2}\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}substitute }{x}_1=0,x_2=-2,y_1=-5,y_2=0\\ =\sqrt{{\left(-2\right)}^2+{\left(5\right)}^2}\\ =\sqrt{4+25}\\ =\sqrt{29}\end{array}$

So, the length of $DA=\sqrt{29} \text{units}$.

Step 4 – Find the area of the square.

The area of a square with side length s is $A=s^2.$

Here, $s=\sqrt{29}\text{ units}$.

Then,

 $\begin{array}{l}A={\left(\sqrt{29}\text{ units}\right)}^2\\ A=29{\text{ units}}^2\end{array}$

Step 5– Check whether the option A is correct or incorrect.

The area of the given square $ABCD$ is $29{\text{ units}}^2$.

This area value is not equal to the value in the option A.

So, the option A is incorrect.

Step 6 – Check whether the option B is correct or incorrect.

The area of the given square ABCD is $29{\text{ units}}^2$.

This area value is not equal to the value in the option B.

So, the option B is incorrect.

Step 7 – Check whether the option C is correct or incorrect.

The area of the given square $ABCD$ is $29{\text{ units}}^2$.

This area value is equal to the value in the option C.

So, the option C is correct.

Step 8 – Check whether the option D is correct or incorrect.

The area of the given square ABCD is $29{\text{ units}}^2$.

This area value is not equal to the value in the option D.

So, the option D is incorrect.

The correct option is C, because the area of the square $ABCD$ is $29{\text{ unit}}^2$.