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Construct the triangle ABC such that |AB| = 8 cm, |BC| = 5 cm, |AC| = 5 cm - Leaving Cert Mathematics - Question Question 1 - 2013
Question Question 1
Construct the triangle ABC such that |AB| = 8 cm, |BC| = 5 cm, |AC| = 5 cm. The point A is given to you. (b) On the same diagram, construct the image of the triangl... show full transcript
Worked Solution & Example Answer:Construct the triangle ABC such that |AB| = 8 cm, |BC| = 5 cm, |AC| = 5 cm - Leaving Cert Mathematics - Question Question 1 - 2013
Step 1
Construct the triangle ABC such that |AB| = 8 cm, |BC| = 5 cm, |AC| = 5 cm.
Answer
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Position Point A: Begin by placing point A on your plane.
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Draw Segment AB: From point A, use a ruler to measure and draw a line segment AB that is 8 cm long.
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Draw Arc for Point C: With the compass, set the width to 5 cm (the length of AC) and place the compass point on A to draw an arc.
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Mark Point B: Place the compass on point B. Then, set it to 5 cm (the length of BC) and draw another arc intersecting with the first arc drawn from A. This intersection determines point C.
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Connect Points: Finally, draw the segments AC and BC to complete triangle ABC.
Step 2
On the same diagram, construct the image of the triangle ABC under the axial symmetry in AB.
Answer
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Identify Line AB: Ensure line AB is clearly marked, as it will serve as the axis of symmetry.
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Locate Point C: Identify the location of point C from triangle ABC.
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Reflect Point C: To find the image C’, measure the perpendicular distance from C to line AB, then extend this distance on the opposite side of line AB to mark point C’.
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Mark C’: Draw line segment AC’ and BC’ to connect the new point C’ to points A and B, thus creating triangle A’B’C’ on the opposite side of line AB.
Step 3
Justify the statement 'AC’BC is a parallelogram'.
Answer
To justify that AC’BC is a parallelogram, we show:
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Opposite Sides: In triangles ABC and AC’B, the sections AC and C’B are equal due to reflection over line AB.
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Diagonals: The diagonals [AB] and [C’C] bisect each other, as point C is reflected directly across line AB, therefore both segments meet at their midpoint.
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Conclusion: Since both pairs of opposite sides are equal and the diagonals bisect each other, AC’BC qualifies as a parallelogram.