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Construct and label the orthocentre of the triangle ABC in the diagram below - Leaving Cert Mathematics - Question 5 - 2019
Question 5
Construct and label the orthocentre of the triangle ABC in the diagram below. Show any construction lines or arcs clearly. In the diagram below O is the centre of c... show full transcript
Worked Solution & Example Answer:Construct and label the orthocentre of the triangle ABC in the diagram below - Leaving Cert Mathematics - Question 5 - 2019
Step 1
Construct and label the orthocentre of the triangle ABC in the diagram below. Show any construction lines or arcs clearly.
Answer
To construct the orthocentre of triangle ABC:
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Draw the triangle: Start with points A, B, and C. Connect these points to form triangle ABC.
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Construct the altitudes: Drop a perpendicular from vertex A to line BC and label this intersection point as D. Repeat this for vertices B and C to lines AC and AB, labeling the intersection points as E and F respectively.
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Locate the orthocentre: The point where all three altitudes intersect is the orthocentre H. Mark this point clearly on the diagram.
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Show construction lines: Ensure that all altitude lines AD, BE, and CF are drawn as dashed lines to indicate they are construction lines.
Step 2
Find, with justification, |ABE|.
Answer
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Given: |CD| = |OB| (radius of the circle).
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Establish relationships: Since [AB] is a diameter, and BE is a tangent to the circle at B, angle ABE is a right angle by the tangent-chord theorem, thus:
[ |ABE| = 90^\circ ]
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Calculate internal angles: Triangle AOD is equilateral (as all sides are equal), leading to:
[ |AOD| = \frac{120^\circ}{2} = 60^\circ ]
[ |ADO| = 60^\circ \text{ (by Alternate angles property)} ] -
Use the properties of angles in triangles: With AOD being isosceles, angles AOD is equal to ADO, thus confirming:
[ |ABE| = 180^\circ - 90^\circ - 60^\circ = 30^\circ ]
Hence, the measure of angle |ABE| is [ 30^\circ ].