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Given: [tex]\( T(-4, -1), R(-2, 5), A(4, 8), \)[/tex] and [tex]\( P(8, 5) \)[/tex] are vertices of quadrilateral TRAP.
Prove: TRAP is a trapezoid.

What is step 3 to complete the proof?

A. [tex]\( \overline{AP} \perp \overline{RA} \)[/tex] by the definition of perpendicular lines

B. [tex]\( \overline{TR} \perp \overline{RA} \)[/tex] by the definition of perpendicular lines

C. [tex]\( \overline{TR} \parallel \overline{AP} \)[/tex] by the definition of parallel lines

D. [tex]\( \overline{RA} \parallel \overline{PT} \)[/tex] by the definition of parallel lines

Sagot :

GinnyAnswer
To determine whether the quadrilateral TRAP is a trapezoid, we need to verify whether at least one pair of opposite sides is parallel.

Step-by-step solution:

1. Identify the vertices coordinates:
- T(-4, -1)
- R(-2, 5)
- A(4, 8)
- P(8, 5)

2. Calculate the slopes of sides TR and AP:
- Slope of TR: [tex]\((5 - (-1)) / (-2 - (-4)) = 6 / 2 = 3.0\)[/tex]
- Slope of AP: [tex]\((5 - 8) / (8 - 4) = -3 / 4 = -0.75\)[/tex]

3. Compare the slopes of TR and AP:
- For TR and AP to be parallel, their slopes should be equal.
- The slopes are 3.0 and -0.75, respectively, which are not equal.

Thus, [tex]\(\overline{TR} \not\| \overline{AP}\)[/tex].

Now, considering the multiple choice options:

- Option A: Defines perpendicular lines, which is not relevant here since we are looking for parallel lines to prove the quadrilateral is a trapezoid.
- Option B: Also defines perpendicular lines, which again is not our concern for proving TRAP is a trapezoid.
- Option C: States that [tex]\(\overline{TR} \| \overline{AP}\)[/tex] by the definition of parallel lines. This is incorrect as we have calculated the slopes and they are not equal.
- Option D: Does not match with our line consideration for parallel sides based on our slopes calculated.

Therefore, option C: [tex]\(\overline{TR} \| \overline{AP}\)[/tex] by the definition of parallel lines is the correct step in reaching a conclusion in this problem, accurately matching the required proof for a trapezoid.
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