In the acute-angled triangle ABC
AP ⊥ BC, BQ ⊥ AC and CR ⊥ AB - Leaving Cert Mathematics - Question 6B - 2013
Question 6B
In the acute-angled triangle ABC
AP ⊥ BC, BQ ⊥ AC and CR ⊥ AB.
Prove that
|∠ABQ| + |∠BCR| + |∠CAP| = 90°.
Worked Solution & Example Answer:In the acute-angled triangle ABC
AP ⊥ BC, BQ ⊥ AC and CR ⊥ AB - Leaving Cert Mathematics - Question 6B - 2013
Step 1
In the triangle APC, show that |∠CAP| + |∠APC| = 90°.
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Answer
From triangle APC, by definition of perpendicular lines, we have |∠CAP| + |∠APC| = 90°.
Step 2
In the triangle QBC, show that |∠BQC| + |∠BCQ| = 90°.
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Answer
In triangle QBC, since BQ ⊥ AC, it follows that |∠BQC| + |∠BCQ| = 90°.
Step 3
Use the results from the previous triangles to show that |∠ABQ| + |∠BCR| + |∠CAP| = 90°.
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Answer
In triangle RBC, we can establish: |∠RBC| + |∠BRK| + |∠BKR| = 90° due to the properties of triangle angles summing up to 180°. By substituting the previous results, we find that |∠ABQ| + |∠BCR| + |∠CAP| = 90°.
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