Question

1. r is between q and t and r, q and t are collinear. qr = 3x - 2, rt = 6x + 7, qt = 13x - 3. find qr, rt and qt.

Explanation:

Step1: Use segment - addition postulate

Since $R$ is between $Q$ and $T$ and they are collinear, $QR + RT=QT$. Substitute the given expressions: $(3x - 2)+(8x + 7)=13x-3$.

Step2: Simplify the left - hand side

Combine like terms: $3x-2 + 8x+7=(3x + 8x)+(-2 + 7)=11x + 5$. So, $11x + 5=13x-3$.

Step3: Solve for $x$

Subtract $11x$ from both sides: $11x+5-11x=13x - 3-11x$, which gives $5 = 2x-3$. Then add 3 to both sides: $5 + 3=2x-3+3$, so $8 = 2x$. Divide both sides by 2: $x = 4$.

Step4: Find $QR$

Substitute $x = 4$ into the expression for $QR$: $QR=3x-2=3\times4-2=12 - 2=10$.

Step5: Find $RT$

Substitute $x = 4$ into the expression for $RT$: $RT=8x + 7=8\times4+7=32 + 7=39$.

Step6: Find $QT$

Substitute $x = 4$ into the expression for $QT$: $QT=13x-3=13\times4-3=52-3=49$.

Answer:

$QR = 10$, $RT = 39$, $QT = 49$