Question

the coordinates of midpoint \\(m\\) and endpoint \\(e\\) of a segment are \\(m(35, -3)\\) and \\(e(15, 26)\\). students were asked to find two equations that could be used to find the coordinates of the other endpoint, \\(f(x_2, y_2)\\).

josiah set up the equations as follows:

josiahs work
\\(\frac{15 + x_2}{2} = -3\\), \\(\frac{26 + y_2}{2} = 35\\)

which statement best describes josiahs work?

• the 15 and 26 should be switched.
• the equations are correct.
• the equations should be \\(\frac{35 + x_2}{2} = 15\\), \\(\frac{-3 + y_2}{2} = 26\\)
• the -3 and 35 should be switched.

Explanation:

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</pre_analysis>

<reasoning>

Identify the given values and formulas

We are given:

• Midpoint \(M(x_m, y_m) = (35, -3)\)
• Endpoint \(E(x_1, y_1) = (15, 26)\)
• Other endpoint \(F(x_2, y_2)\)

The midpoint formula states:
\[
x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}
\]

Set up the correct equations

Substitute the given coordinates into the midpoint formulas:
\[
35 = \frac{15 + x_2}{2}
\]
\[
-3 = \frac{26 + y_2}{2}
\]

Analyze Josiah's work

Josiah set up the equations as:
\[
\frac{15 + x_2}{2} = -3, \quad \frac{26 + y_2}{2} = 35
\]

Comparing Josiah's equations to the correct equations:

• In his first equation, he set the \(x\)-coordinate average equal to the \(y\)-coordinate of the midpoint \(-3\).
• In his second equation, he set the \(y\)-coordinate average equal to the \(x\)-coordinate of the midpoint \(35\).

Therefore, the values \(-3\) and \(35\) are switched.

Evaluate the given options

• Option 1: "The 15 and 26 should be switched" (Incorrect, these are the correct endpoint coordinates paired with their respective variables).
• Option 2: "The equations are correct" (Incorrect).
• Option 3: "The equations should be \(\frac{35 + x_2}{2} = 15, \frac{-3 + y_2}{2} = 26\)" (Incorrect, this treats \(M\) as an endpoint and \(E\) as the midpoint).
• Option 4: "The \(-3\) and \(35\) should be switched" (Correct, switching them yields the correct system of equations).

</reasoning>

<answer>
<mcq-option>The 15 and 26 should be switched</mcq-option>
<mcq-option>The equations are correct</mcq-option>
<mcq-option>The equations should be \(\frac{35 + x_2}{2} = 15, \frac{-3 + y_2}{2} = 26\)</mcq-option>
<mcq-correct>The \(-3\) and \(35\) should be switched</mcq-correct>
</answer>

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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Midpoint Formula"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Identify the given values and formulas

We are given:

• Midpoint \(M(x_m, y_m) = (35, -3)\)
• Endpoint \(E(x_1, y_1) = (15, 26)\)
• Other endpoint \(F(x_2, y_2)\)

The midpoint formula states:
\[
x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}
\]

Set up the correct equations

Substitute the given coordinates into the midpoint formulas:
\[
35 = \frac{15 + x_2}{2}
\]
\[
-3 = \frac{26 + y_2}{2}
\]

Analyze Josiah's work

Josiah set up the equations as:
\[
\frac{15 + x_2}{2} = -3, \quad \frac{26 + y_2}{2} = 35
\]

Comparing Josiah's equations to the correct equations:

• In his first equation, he set the \(x\)-coordinate average equal to the \(y\)-coordinate of the midpoint \(-3\).
• In his second equation, he set the \(y\)-coordinate average equal to the \(x\)-coordinate of the midpoint \(35\).

Therefore, the values \(-3\) and \(35\) are switched.

Evaluate the given options

• Option 1: "The 15 and 26 should be switched" (Incorrect, these are the correct endpoint coordinates paired with their respective variables).
• Option 2: "The equations are correct" (Incorrect).
• Option 3: "The equations should be \(\frac{35 + x_2}{2} = 15, \frac{-3 + y_2}{2} = 26\)" (Incorrect, this treats \(M\) as an endpoint and \(E\) as the midpoint).
• Option 4: "The \(-3\) and \(35\) should be switched" (Correct, switching them yields the correct system of equations).

</reasoning>

<answer>
<mcq-option>The 15 and 26 should be switched</mcq-option>
<mcq-option>The equations are correct</mcq-option>
<mcq-option>The equations should be \(\frac{35 + x_2}{2} = 15, \frac{-3 + y_2}{2} = 26\)</mcq-option>
<mcq-correct>The \(-3\) and \(35\) should be switched</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Midpoint Formula"
]
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