Question

in triangle pmt, $gh = x + 3$, $pg = 4x + 1$, and $gj = 2x - 1$
what is the length of segment pg?
$x=$ type your answer...
$pg=$ type your answer...
17 fill in the blank - 2 points.
do the following points create a parallelogram: k(-3,2), l(4,2), n(-4,-2), m(3,-2)?
no because opposite slopes are the sa and opposite sides are congru

Explanation:

Step1: Recognize centroid property

In a triangle, the centroid divides each median into a ratio of $2:1$, so the longer segment (from vertex to centroid) is twice the shorter segment (from centroid to side). Here, $PG = 2 \times GJ$.

Step2: Substitute given expressions

Replace $PG$ with $4x+1$ and $GJ$ with $2x-1$:
$4x + 1 = 2(2x - 1)$

Step3: Simplify and solve for $x$

Expand the right side:
$4x + 1 = 4x - 2$
Subtract $4x$ from both sides:
$1 = -2$
This is a contradiction, which means we use the other segment pair: $PG = 2 \times GH$
Substitute values:
$4x + 1 = 2(x + 3)$
Expand right side:
$4x + 1 = 2x + 6$
Subtract $2x$ from both sides:
$2x + 1 = 6$
Subtract 1 from both sides:
$2x = 5$
Solve for $x$:
$x = \frac{5}{2} = 2.5$

Step4: Calculate length of $PG$

Substitute $x=2.5$ into $PG=4x+1$:
$PG = 4(2.5) + 1$

A parallelogram has opposite sides that are parallel (equal slopes) and congruent. Calculate slopes of sides:

• Slope of $KL$: $\frac{2-2}{4-(-3)} = 0$
• Slope of $NM$: $\frac{-2-(-2)}{3-(-4)} = 0$
• Slope of $KN$: $\frac{-2-2}{-4-(-3)} = 4$
• Slope of $LM$: $\frac{-2-2}{3-4} = 4$

Opposite slopes are equal, so opposite sides are parallel. Calculate side lengths:

• Length of $KL$: $\sqrt{(4-(-3))^2 + (2-2)^2} = 7$
• Length of $NM$: $\sqrt{(3-(-4))^2 + (-2-(-2))^2} = 7$
• Length of $KN$: $\sqrt{(-4-(-3))^2 + (-2-2)^2} = \sqrt{17}$
• Length of $LM$: $\sqrt{(3-4)^2 + (-2-2)^2} = \sqrt{17}$

Opposite sides are congruent, so the points form a parallelogram.

Answer:

$x=2.5$
$PG=11$

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For the parallelogram question: