Question

line segment pr is graphed below. if pr is one side of parallelogram prst, which coordinate pairs could represent vertices s and t? (0,0) and (4,1) (7,2) and (5,11) (9,2) and (6,10) (10,11) and (7,9)

Explanation:

Step1: Find the vector of \(PR\)

For points \(P(-7,4)\) and \(R(-4,-4)\), the vector \(\overrightarrow{PR}\) is \((-4 - (-7),-4 - 4)=(3,-8)\).
In a parallelogram \(PRST\), \(\overrightarrow{PS}=\overrightarrow{RT}\) and \(\overrightarrow{PR}=\overrightarrow{ST}\).
Let's check each option:

Option 1: \((6,9)\) and \((4,1)\)

The vector between \((6,9)\) and \((4,1)\) is \((4 - 6,1 - 9)=(-2,-8)
eq(3,-8)\)

Option 2: \((7,3)\) and \((5,11)\)

The vector between \((7,3)\) and \((5,11)\) is \((5 - 7,11 - 3)=(-2,8)
eq(3,-8)\)

Option 3: \((9,2)\) and \((6,10)\)

The vector between \((9,2)\) and \((6,10)\) is \((6 - 9,10 - 2)=(-3,8)
eq(3,-8)\)

Option 4: \((10,11)\) and \((7,3)\)

The vector between \((10,11)\) and \((7,3)\) is \((7 - 10,3 - 11)=(-3,-8)\).
If we consider the translation, if we assume \(P(-7,4)\) and \(R(-4,-4)\), if we add the vector \((3,-8)\) to \(P(-7,4)\) we get \(S(-7+3,4 - 8)=(-4,-4)\) (which is \(R\)), if we consider another way.
Let's use the mid - point formula. The mid - point of \(PR\) is \((\frac{-7-4}{2},\frac{4-4}{2})=(-\frac{11}{2},0)\).
For a parallelogram, the mid - point of \(ST\) is the same as the mid - point of \(PR\).
For points \(S(10,11)\) and \(T(7,3)\), the mid - point is \((\frac{10 + 7}{2},\frac{11+3}{2})=(\frac{17}{2},7)\) (incorrect).
Let's use the property of parallel sides.
The slope of \(PR\) is \(m_{PR}=\frac{-4 - 4}{-4+7}=\frac{-8}{3}\)
For a parallelogram, opposite sides are parallel.
Let's assume \(P(-7,4)\), if we consider the translation.
If we move from \(P(-7,4)\) to \(R(-4,-4)\) (right \(3\) units and down \(8\) units).
If we assume \(S(x_1,y_1)\) and \(T(x_2,y_2)\)
Another way:
The vector approach: In a parallelogram \(PRST\), \(\overrightarrow{PR}=\overrightarrow{ST}\)
Let \(P(-7,4)\), \(R(-4,-4)\)
If \(S(10,11)\) and \(T(7,3)\)
\(\overrightarrow{ST}=(7 - 10,3 - 11)=(-3,-8)\)
\(\overrightarrow{PR}=(-4+7,-4 - 4)=(3,-8)\) (magnitude is same, direction is opposite, but in a parallelogram \(PR\parallel ST\) and \(PS\parallel RT\))

Answer:

\((10,11)\) and \((7,3)\)