Question

one endpoint of a line segment is located at (-2, 2). the other endpoint is 14 units right and the y-coordinate of the midpoint of the segment is 8. write the ordered pair that represents the other end of the segment.

Explanation:

Step1: Analyze x - coordinate change

The first endpoint has an x - coordinate of \(-2\). The other endpoint is 14 units to the right. To find the new x - coordinate, we use the formula for horizontal translation: \(x_{new}=x_{old}+14\). Substituting \(x_{old}=-2\), we get \(x_{new}=-2 + 14=12\).

Step2: Analyze y - coordinate change

The mid - point's y - coordinate is 8. Wait, actually, the problem says "the y - coordinate of the midpoint of the segment is 8"? No, re - reading: "the y - coordinate of the midpoint of the segment is 8"? Wait, no, the problem says "the other endpoint is 14 units right and the y - coordinate of the midpoint of the segment is 8". Wait, no, let's re - parse the problem: One endpoint of a line segment is at \((-2,2)\). The other endpoint is 14 units right (so change in x: \(x_2=x_1 + 14\), where \(x_1=-2\)) and the y - coordinate of the midpoint of the segment is 8. Wait, no, maybe I misread. Wait, the problem says: "The other endpoint is 14 units right and the y - coordinate of the midpoint of the segment is 8. Write the ordered pair that represents the other end of the segment."

Wait, let's use the mid - point formula. Let the first endpoint be \(A(-2,2)\) and the second endpoint be \(B(x,y)\). The mid - point \(M\) has coordinates \((\frac{-2 + x}{2},\frac{2 + y}{2})\). We know that the horizontal distance between \(A\) and \(B\) is 14 units. Since it's 14 units to the right, \(x-(-2)=14\) (because moving right increases the x - coordinate), so \(x + 2=14\), so \(x=14 - 2 = 12\).

Now, the y - coordinate of the mid - point is 8. So using the mid - point formula for y - coordinate: \(\frac{2 + y}{2}=8\). Multiply both sides by 2: \(2 + y=16\). Subtract 2 from both sides: \(y = 16-2=14\). Wait, but that seems off. Wait, maybe the problem is that the "14 units right" is the horizontal component, and we need to find the y - coordinate. Wait, maybe I misread the problem. Let's re - read: "One endpoint of a line segment is located at \((-2,2)\). The other endpoint is 14 units right and the y - coordinate of the midpoint of the segment is 8. Write the ordered pair that represents the other end of the segment."

Wait, horizontal displacement: from \(x=-2\) to \(x\) is 14 units right, so \(x=-2 + 14 = 12\). Now, mid - point y - coordinate is 8. Let the other endpoint be \((12,y)\). Mid - point y - coordinate: \(\frac{2 + y}{2}=8\). Solving for \(y\): \(2 + y=16\), so \(y = 14\). Wait, but maybe the problem is that the length of the segment is related? Wait, no, the problem says "the y - coordinate of the midpoint of the segment is 8". So using mid - point formula:

Mid - point \(M=(\frac{-2 + x}{2},\frac{2 + y}{2})\)

We know that the horizontal distance between the two endpoints is 14, so \(x-(-2)=14\) (since it's 14 units to the right), so \(x = 12\).

For the y - coordinate of the mid - point: \(\frac{2 + y}{2}=8\)

Multiply both sides by 2: \(2 + y=16\)

Subtract 2: \(y = 14\)

Wait, but let's check. If \(A(-2,2)\) and \(B(12,14)\), the mid - point is \((\frac{-2+12}{2},\frac{2 + 14}{2})=(5,8)\), which has a y - coordinate of 8. And the horizontal distance between \(-2\) and 12 is \(12-(-2)=14\), which matches the "14 units right" condition. So the other endpoint is \((12,14)\).

Wait, maybe I made a mistake earlier. Let's go step by step:

1. Find the x - coordinate of the other endpoint:
The first endpoint has \(x=-2\). Moving 14 units to the right means adding 14 to the x - coordinate. So \(x=-2 + 14=12\).

2. Find the y - coordinate of the other endpoint:
Let the two endpoints be \((-2,2)\) and \((12,y)\). The mid - point formula for the y - coordinate is \(\frac{2 + y}{2}\). We know the mid - point's y - coordinate is 8. So:

\(\frac{2 + y}{2}=8\)

Multiply both sides by 2: \(2 + y = 16\)

Subtract 2 from both sides: \(y=16 - 2=14\)

Answer:

\((12,14)\)