finding the line segment?

Question

let $a$ and $b$ belong $\mathbb R$. If $P(x,y)$ is the point in the plane, define $f(P)= ax + by$. let the line segment $AB$ bisect the line segment $CD$.If $f(A)=5$ , $f(B)=5$ and $f(C)= 10$ , find $f(D).$

I was trying this question that AB bisect the CD that mean it mean they have equal sides and their mid point have same coordinates. But my answer is f(A)= f(B)=5 and f(C)= f(D) = 10 and my answer is f(D)=10 Is my answer is correct or not, im not sure i have only used my own logic

if anbody help me i would be very thankful to him

Answer 1

$f(D)=0$. I assume the line segments bisect each other.

Given: $$f(A)=aa_1+ba_2=5$$ $$f(B)=ab_1+bb_2=5$$ $$f(C)=ac_1+bc_2=10$$ $$f(D)=ad_1+bd_2=?$$ Note: $$\frac{f(A)+f(B)}{2}=5$$ $$\frac{f(C)+f(D)}{2}=\frac{f(A)+f(B)}{2} \Rightarrow \frac{10+f(D)}{2}=5 \Rightarrow f(D)=0.$$

Answer 2

An equation of $AB$ it's $ax+by=5$.

Thus, $$a\cdot\frac{x_C+x_D}{2}+b\cdot\frac{y_C+y_D}{2}=5$$ or $$10+f(D)=10$$ or $$f(D)=0$$.