Question

if line segment ru is considered the base of parallelogram rstu, what is the corresponding height of the parallelogram?

4.5 units
5.4 units
9.0 units
10.8 units

Explanation:

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<reasoning>

Identify the coordinates of the vertices

Find the coordinates of the vertices of parallelogram \(RSTU\) from the grid:

• \(R = (1, 1)\)
• \(S = (7, 0)\)
• \(T = (10, 4)\)
• \(U = (4, 5)\)

Calculate the area of the parallelogram

Using the Shoelace Formula knowledge point
\[

$$\begin{aligned} \text{Area} &= \frac{1}{2} |(1\cdot0 + 7\cdot4 + 10\cdot5 + 4\cdot1) - (1\cdot7 + 0\cdot10 + 4\cdot4 + 5\cdot1)| \\ &= \frac{1}{2} |(0 + 28 + 50 + 4) - (7 + 0 + 16 + 5)| \\ &= \frac{1}{2} |82 - 28| \\ &= \frac{1}{2} \cdot 54 \\ &= 27 \end{aligned}$$

\]

Calculate the length of the base RU

Using the Area of Polygons knowledge point
\[

$$\begin{aligned} RU &= \sqrt{(4 - 1)^2 + (5 - 1)^2} \\ &= \sqrt{3^2 + 4^2} \\ &= \sqrt{9 + 16} \\ &= 5 \end{aligned}$$

\]

Calculate the corresponding height

Using the Area of Polygons knowledge point
\[

$$\begin{aligned} \text{Area} &= \text{base} \times \text{height} \\ 27 &= 5 \times h \\ h &= \frac{27}{5} \\ h &= 5.4 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 4.5 units</mcq-option>
<mcq-correct>(B) 5.4 units</mcq-correct>
<mcq-option>(C) 9.0 units</mcq-option>
<mcq-option>(D) 10.8 units</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[1, 1]],
"properties": {
"name": "R(1,1)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[7, 0]],
"properties": {
"name": "S(7,0)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[10, 4]],
"properties": {
"name": "T(10,4)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 5]],
"properties": {
"name": "U(4,5)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "polygon",
"params": [[1, 1], [7, 0], [10, 4], [4, 5]],
"properties": {
"fillColor": "rgba(140, 85, 242, 0.15)",
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "segment",
"params": [[4, 5], [1.24, 1.32]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 2,
"dash": 2,
"name": "h = 5.4",
"withLabel": true
}
}
]
}
</plot>

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"question_type": "Multiple Choice",
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"Geometry",
"Parallelogram Height in Coordinate Plane"
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Answer:

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<reasoning>

Identify the coordinates of the vertices

Find the coordinates of the vertices of parallelogram \(RSTU\) from the grid:

• \(R = (1, 1)\)
• \(S = (7, 0)\)
• \(T = (10, 4)\)
• \(U = (4, 5)\)

Calculate the area of the parallelogram

Using the Shoelace Formula knowledge point
\[

$$\begin{aligned} \text{Area} &= \frac{1}{2} |(1\cdot0 + 7\cdot4 + 10\cdot5 + 4\cdot1) - (1\cdot7 + 0\cdot10 + 4\cdot4 + 5\cdot1)| \\ &= \frac{1}{2} |(0 + 28 + 50 + 4) - (7 + 0 + 16 + 5)| \\ &= \frac{1}{2} |82 - 28| \\ &= \frac{1}{2} \cdot 54 \\ &= 27 \end{aligned}$$

\]

Calculate the length of the base RU

Using the Area of Polygons knowledge point
\[

$$\begin{aligned} RU &= \sqrt{(4 - 1)^2 + (5 - 1)^2} \\ &= \sqrt{3^2 + 4^2} \\ &= \sqrt{9 + 16} \\ &= 5 \end{aligned}$$

\]

Calculate the corresponding height

Using the Area of Polygons knowledge point
\[

$$\begin{aligned} \text{Area} &= \text{base} \times \text{height} \\ 27 &= 5 \times h \\ h &= \frac{27}{5} \\ h &= 5.4 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 4.5 units</mcq-option>
<mcq-correct>(B) 5.4 units</mcq-correct>
<mcq-option>(C) 9.0 units</mcq-option>
<mcq-option>(D) 10.8 units</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[1, 1]],
"properties": {
"name": "R(1,1)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[7, 0]],
"properties": {
"name": "S(7,0)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[10, 4]],
"properties": {
"name": "T(10,4)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 5]],
"properties": {
"name": "U(4,5)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "polygon",
"params": [[1, 1], [7, 0], [10, 4], [4, 5]],
"properties": {
"fillColor": "rgba(140, 85, 242, 0.15)",
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "segment",
"params": [[4, 5], [1.24, 1.32]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 2,
"dash": 2,
"name": "h = 5.4",
"withLabel": true
}
}
]
}
</plot>

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