Question

1. a line segment on the coordinate plane has endpoints (2,4) and (4,y). the midpoint of the segment is point (3,7). what is the value of y?

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Given endpoints \((x_1,y_1)=(2,4)\) and \((x_2,y_2)=(4,y)\) and mid - point \((x_m,y_m)=(3,7)\).

Step2: Use x - coordinate of mid - point formula

For the x - coordinate of the mid - point, \(\frac{x_1 + x_2}{2}=x_m\). Substitute \(x_1 = 2\), \(x_2 = 4\) and \(x_m=3\). \(\frac{2 + 4}{2}=\frac{6}{2}=3\) (which is correct as per the given mid - point x - coordinate).

Step3: Use y - coordinate of mid - point formula

For the y - coordinate of the mid - point, \(\frac{y_1 + y_2}{2}=y_m\). Substitute \(y_1 = 4\), \(y_2 = y\) and \(y_m = 7\). We get \(\frac{4 + y}{2}=7\).

Step4: Solve for y

Multiply both sides of the equation \(\frac{4 + y}{2}=7\) by 2: \(4 + y=14\). Then subtract 4 from both sides: \(y=14 - 4=10\).

Answer:

10