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Construct the centroid of the triangle ABC below - Leaving Cert Mathematics - Question 6 - 2015
Question 6
Construct the centroid of the triangle ABC below. Show all construction lines. (Where measurement is used, show all relevant measurements and calculations clearly.) ... show full transcript
Worked Solution & Example Answer:Construct the centroid of the triangle ABC below - Leaving Cert Mathematics - Question 6 - 2015
Step 1
Construct the centroid of the triangle ABC below. Show all construction lines.
Answer
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Construct sides of the triangle: Draw triangle ABC with | AC | = 11.1 cm and | BC | = 11.7 cm.
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Locate midpoints: Find midpoints D and E of sides AC and BC respectively.
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Draw medians: Draw line segments AD and BE from vertices A to midpoint D and B to midpoint E.
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Determine intersection: The point where medians AD and BE intersect is the centroid G of triangle ABC.
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Show measurements clearly: Clearly label all measurements and significant points in the diagram.
Step 2
Prove that, if three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal line.
Answer
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Given: AD || BE || CF, with | AB | = | BC |.
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Construction: Draw AE' || DE, which intersects EB at E' and CF at F'. Then, draw FB' || AB, which intersects EB at B'.
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Proof:
- Since | BF' | = | BC | (opposite sides in a parallelogram),
- And | AB' | = | AB | (by assumption),
- We have | AE' | = | EF' | = | DE |.
- Thus, by the properties of parallel lines and transversals, we conclude that | DE | = | EF |.