$$ 14 \begin{array}{l}A B C D ext { is a square where } A(0,0), B(3, y) ext {, if its side length is } 5 ext { un } \ ext { find the positive value of } y\end{array} $$

Question

$$ 14 \begin{array}{l}A B C D 	ext { is a square where } A(0,0), B(3, y) 	ext {, if its side length is } 5 	ext { un } \ 	ext { find the positive value of } y\end{array} $$

UpStudy AI · Accepted Answer

To determine the positive value of $$ y $$ for the square $$ ABCD $$ with vertices $$ A(0,0) $$ and $$ B(3, y) $$, given that the side length is $$ 5 $$ units, follow these steps:

1. **Understanding the Problem:**
   
   - $$ A(0,0) $$ and $$ B(3, y) $$ are consecutive vertices of the square.
   - The distance between $$ A $$ and $$ B $$ is equal to the side length of the square, which is $$ 5 $$ units.

2. **Using the Distance Formula:**
   
   The distance $$ AB $$ between points $$ A(x_1, y_1) $$ and $$ B(x_2, y_2) $$ is given by:
   $$
   AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   $$
   
   Plugging in the coordinates of $$ A $$ and $$ B $$:
   $$
   5 = \sqrt{(3 - 0)^2 + (y - 0)^2}
   $$
   $$
   5 = \sqrt{9 + y^2}
   $$

3. **Solving for $$ y $$:**
   
   Square both sides to eliminate the square root:
   $$
   25 = 9 + y^2
   $$
   $$
   y^2 = 25 - 9
   $$
   $$
   y^2 = 16
   $$
   $$
   y = \pm 4
   $$
   
   Since we're asked for the positive value:
   $$
   y = 4
   $$

**Answer:** The positive value of $$ y $$ is $$ 4 $$.
The positive value of $$ y $$ is $$ 4 $$.

\( 14 \begin{array}{l}A B C D \text { is a square where } A(0,0), B(3, y) \text {, if its side length is } 5 \text { un } \\ \text { find the positive value of } y\end{array} \)

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