if v is the mid - point of $overline{qs}$ and w is the mid - point of $overline{rs}$, then what is vs?\n4…

if v is the mid - point of $overline{qs}$ and w is the mid - point of $overline{rs}$, then what is vs?\n4 units\n8 units\n10 units\n20 units

if v is the mid - point of $overline{qs}$ and w is the mid - point of $overline{rs}$, then what is vs?\n4 units\n8 units\n10 units\n20 units

Answer

Explanation:

Step1: Use the mid - segment theorem

In a triangle, the segment joining the mid - points of two sides is parallel to the third side and half its length. So, $VW=\frac{1}{2}QR$. We have $VW = 2a - 2$ and $QR=3a + 6$. Then $2a-2=\frac{1}{2}(3a + 6)$.

Step2: Solve the equation for $a$

Multiply both sides of the equation $2a-2=\frac{1}{2}(3a + 6)$ by 2 to get $2(2a - 2)=3a+6$. Expand the left - hand side: $4a-4 = 3a + 6$. Subtract $3a$ from both sides: $4a-3a-4=3a-3a + 6$, which simplifies to $a-4 = 6$. Add 4 to both sides: $a=6 + 4=10$.

Step3: Find the length of $VS$

Since $V$ is the mid - point of $QS$ and $QS = 2a$, substituting $a = 10$ into $QS = 2a$, we get $QS=2\times10 = 20$. Then $VS=\frac{1}{2}QS$. So $VS = 10$ units.

Answer:

10 units