Question

in the diagram below of triangle nop, q is the mid - point of np and r is the mid - point of op. qr=-23 + 8x, and no=-3x + 49, what is the measure of qr?

Explanation:

Step1: Apply mid - segment theorem

In a triangle, the line segment joining the mid - points of two sides is half the length of the third side. So, $QR=\frac{1}{2}NO$.

Step2: Set up the equation

We have $- 23 + 8x=\frac{1}{2}(-3x + 49)$.
Multiply both sides of the equation by 2 to get rid of the fraction: $2(-23 + 8x)=-3x + 49$.
Expand the left - hand side: $-46+16x=-3x + 49$.

Step3: Solve for x

Add $3x$ to both sides: $-46 + 16x+3x=-3x + 49+3x$, which simplifies to $-46 + 19x=49$.
Add 46 to both sides: $-46+19x + 46=49 + 46$, so $19x = 95$.
Divide both sides by 19: $x=\frac{95}{19}=5$.

Step4: Find the length of QR

Substitute $x = 5$ into the expression for $QR$: $QR=-23+8x$.
$QR=-23 + 8\times5=-23 + 40 = 17$.

Answer:

17