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(a) Construct and label the orthocentre of the triangle ABC in the diagram below - Leaving Cert Mathematics - Question 5 - 2019
Question 5
(a) Construct and label the orthocentre of the triangle ABC in the diagram below. Show any construction lines or arcs clearly. (b) In the diagram below O is the cen... show full transcript
Worked Solution & Example Answer:(a) 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.
Answer
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Draw Triangle ABC: Begin by sketching triangle ABC on the provided diagram with points A, B, and C labeled accordingly.
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Construct Altitudes: To find the orthocentre, construct the altitudes from each vertex.
- Altitude from A: Draw a perpendicular line from A to line segment BC.
- Altitude from B: Draw a perpendicular line from B to line segment AC.
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Label the Points of Intersection: Let the intersection of the altitudes from A and B be labeled as point H. This point is the orthocentre of triangle ABC.
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Complete the Construction: Ensure all construction lines are clearly marked, and the orthocentre H is distinctly labeled in the diagram.
Step 2
Find, with justification, |∠BEA|.
Answer
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Identify Given Information: We know that |CD| = rac{1}{2} |AB| and that CD is parallel to AB.
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Triangle Similarity: Since |CD| is half of |AB| and they are parallel, triangle ACD is similar to triangle ABE by the basic proportionality theorem (also known as Thales' theorem).
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Identify Angles: Let |∠ADC| = |∠ABE| since they are corresponding angles.
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Calculate Angles: From the given data, note that:
- |∠ADC| = 60° (as given).
- Therefore, |∠ABE| = 60°.
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Using the Tangent Relationship: Since BE is tangent at B and |∠ABE| is formed with the radius OB, we know that:
- |∠BEA| = 90°.
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Final Calculation: Thus, |∠BEA| = 90° - |∠ABE| = 90° - 60° = 30°.