Final answer:
To make CD perpendicular to EF, we calculate the slope of CD, and then use the negative reciprocal to find the slope for EF. Since the slope for CD is 8/7, the slope for EF is -7/8. Setting the slope formula equal to -7/8 with one of the points on EF and solving for y, we find y to be 3.
Step-by-step explanation:
A student's question involves finding the value of y for point F(4, y) to make line segment CD perpendicular to EF in a coordinate plane. To do this, we need to calculate the slopes of CD and EF and set their product to -1 because the product of the slopes of two perpendicular lines is -1. Calculating slopes of two points requires using the formula (y2 - y1)/(x2 - x1).
For line CD, the given points are C(12, -2) and D(5, -10). The slope of CD (m_CD) is (-10 - (-2))/(5 - 12), which simplifies to -8/-7 or 8/7.
Since slope of EF should be perpendicular to CD, the slope of EF (m_EF) should be the negative reciprocal of CD's slope. So, m_EF is -7/8. Using point E(-4, 10) and F(4, y) to find y, the slope is (y - 10)/(4 - (-4)) which simplifies to (y - 10)/8. Setting this equal to -7/8 (because m_EF = -7/8), we get y - 10 = -7. Solving for y, we find that y equals 3.