Question

aidan and rick are drawing a map of the space used for a game of laser tag. three of the four vertices of the space are shown in the coordinate plane. the fourth vertex is 7 units directly east of vertex a. aidan says that the missing vertex should be at (9, 3). rick says aidan is incorrect. use the drop-down menus to show how rick can explain to aidan why he is incorrect.

Explanation:

To solve this, we first identify the coordinates of vertex \( A \). From the graph, \( A \) is at \( ( - 1,3) \) (assuming the grid lines are 1 unit each, looking at the position relative to the axes).

Step 1: Determine the x - coordinate of the missing vertex

If the fourth vertex is 7 units directly east of vertex \( A \), we use the rule for moving east (which is in the positive \( x \) - direction). The \( x \) - coordinate of \( A \) is \( x_A=- 1 \). To find the \( x \) - coordinate of the point 7 units east of \( A \), we add 7 to the \( x \) - coordinate of \( A \).
The formula for the new \( x \) - coordinate \( x_{new} \) is \( x_{new}=x_A + 7 \)
Substituting \( x_A=-1 \) into the formula, we get \( x_{new}=-1 + 7=6 \)

Step 2: Determine the y - coordinate of the missing vertex

Since the movement is directly east (horizontal movement), the \( y \) - coordinate of the point will be the same as the \( y \) - coordinate of \( A \). The \( y \) - coordinate of \( A \) is \( y_A = 3 \), so the \( y \) - coordinate of the missing vertex is also 3.

Step 3: Compare with Aidan's answer

Aidan says the missing vertex is at \( (9,3) \), but from our calculation, the correct \( x \) - coordinate should be 6 (since \( - 1+7 = 6 \)) and the \( y \) - coordinate is 3. So the correct coordinates of the missing vertex should be \( (6,3) \), not \( (9,3) \).

Answer:

The missing vertex should be at \( (6,3) \) because moving 7 units east from \( A(-1,3) \) means adding 7 to the \( x \) - coordinate (\( - 1+7 = 6 \)) and keeping the \( y \) - coordinate the same (3), so it is not \( (9,3) \).