Question

point a is at $(-8,7.5)$ and point b is at $(6,3.5)$. if point $m = (1 - t)a + tb = (-4.5,6.5)$, what is the value of $t$? you can use the interactive diagram below to help you : diagram showing points a, b, m on a coordinate plane

Explanation:

Step1: Use x - coordinate to set equation

The x - coordinate of \(A\) is \(-8\), the x - coordinate of \(B\) is \(6\), and the x - coordinate of \(M\) is \(-4.5\). From the formula \(M=(1 - t)A + tB\), for the x - coordinates, we have \((1 - t)(-8)+t(6)=-4.5\).
First, expand the left - hand side: \(-8 + 8t+6t=-4.5\).
Combine like terms: \(-8 + 14t=-4.5\).

Step2: Solve for t

Add 8 to both sides of the equation: \(14t=-4.5 + 8\).
Calculate the right - hand side: \(14t = 3.5\).
Divide both sides by 14: \(t=\frac{3.5}{14}\).
Simplify the fraction: \(t = 0.25=\frac{1}{4}\).

We can also verify using the y - coordinate. The y - coordinate of \(A\) is \(7.5\), the y - coordinate of \(B\) is \(3.5\), and the y - coordinate of \(M\) is \(6.5\). From the formula \(M=(1 - t)A + tB\), for the y - coordinates, \((1 - t)(7.5)+t(3.5)=6.5\).
Expand: \(7.5-7.5t + 3.5t=6.5\).
Combine like terms: \(7.5-4t=6.5\).
Subtract 7.5 from both sides: \(-4t=6.5 - 7.5=-1\).
Divide by \(-4\): \(t=\frac{-1}{-4}=\frac{1}{4}=0.25\).

Answer:

\(0.25\) (or \(\frac{1}{4}\))