Question

point a is at (2, 6) and point b is at (-5, -7.5). if point m = (1 - t)a + tb = (-5, -7.5), what is the value of t? you can use the interactive diagram below to help you : diagram with coordinate plane, points, and labels

Explanation:

Step1: Substitute coordinates of A and B

Given \( A=(2,6) \), \( B=(-5,-7.5) \), and \( M=(1 - t)A + tB = (-5,-7.5) \). Substitute the coordinates into the formula for the x - coordinate (we can use either x or y, let's use x first):
The x - coordinate of \( M \) is given by \( (1 - t)x_A + tx_B \). So, \( (1 - t)\times2 + t\times(-5)=-5 \).

Step2: Simplify the equation

Expand the left - hand side: \( 2-2t-5t=-5 \).
Combine like terms: \( 2 - 7t=-5 \).
Subtract 2 from both sides: \( -7t=-5 - 2=-7 \).

Step3: Solve for t

Divide both sides by - 7: \( t=\frac{-7}{-7}=1 \).
We can verify with the y - coordinate. The y - coordinate of \( M \) is \( (1 - t)y_A+ty_B \). Substitute \( t = 1 \), \( y_A = 6 \), \( y_B=-7.5 \):
\( (1 - 1)\times6+1\times(-7.5)=0 - 7.5=-7.5 \), which matches the y - coordinate of \( M \).

Answer:

\( 1 \)