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Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR.Show that triangle ABC and triangle PQR are similar.?
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Sides AB and AC and median AD of a triangle ABC are respectively propo...
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Sides AB and AC and median AD of a triangle ABC are respectively propo...
Given:
- In triangle ABC, sides AB and AC and median AD are respectively proportional to sides PQ and PR and median PM of another triangle PQR.
To prove:
- Triangle ABC and triangle PQR are similar.
Proof:
1. Proportional sides:
- Given that sides AB and AC of triangle ABC are proportional to sides PQ and PR of triangle PQR.
- We can write this as AB/PQ = AC/PR.
2. Proportional medians:
- Given that median AD of triangle ABC is proportional to median PM of triangle PQR.
- We can write this as AD/PM = 1.
3. Similarity criterion:
- Two triangles are similar if their corresponding sides are proportional.
4. Proving similarity:
- Using the proportionality of sides, we can rewrite AB/PQ = AC/PR as AB/AC = PQ/PR.
- Now, we can use the similarity criterion to prove that triangle ABC and triangle PQR are similar.
- By the Angle-Angle (AA) similarity criterion, if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
- In this case, angle BAC of triangle ABC is congruent to angle QPR of triangle PQR (since they are corresponding angles).
- Also, angle ABC of triangle ABC is congruent to angle QRP of triangle PQR (since they are corresponding angles).
- Therefore, triangle ABC and triangle PQR have two pairs of congruent angles, which satisfies the AA similarity criterion.
- Hence, triangle ABC and triangle PQR are similar.
5. Conclusion:
- Triangle ABC and triangle PQR are similar, based on the given conditions and the AA similarity criterion.
- In triangle ABC, sides AB and AC and median AD are respectively proportional to sides PQ and PR and median PM of another triangle PQR.
To prove:
- Triangle ABC and triangle PQR are similar.
Proof:
1. Proportional sides:
- Given that sides AB and AC of triangle ABC are proportional to sides PQ and PR of triangle PQR.
- We can write this as AB/PQ = AC/PR.
2. Proportional medians:
- Given that median AD of triangle ABC is proportional to median PM of triangle PQR.
- We can write this as AD/PM = 1.
3. Similarity criterion:
- Two triangles are similar if their corresponding sides are proportional.
4. Proving similarity:
- Using the proportionality of sides, we can rewrite AB/PQ = AC/PR as AB/AC = PQ/PR.
- Now, we can use the similarity criterion to prove that triangle ABC and triangle PQR are similar.
- By the Angle-Angle (AA) similarity criterion, if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
- In this case, angle BAC of triangle ABC is congruent to angle QPR of triangle PQR (since they are corresponding angles).
- Also, angle ABC of triangle ABC is congruent to angle QRP of triangle PQR (since they are corresponding angles).
- Therefore, triangle ABC and triangle PQR have two pairs of congruent angles, which satisfies the AA similarity criterion.
- Hence, triangle ABC and triangle PQR are similar.
5. Conclusion:
- Triangle ABC and triangle PQR are similar, based on the given conditions and the AA similarity criterion.
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Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR.Show that triangle ABC and triangle PQR are similar.?
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Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR.Show that triangle ABC and triangle PQR are similar.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR.Show that triangle ABC and triangle PQR are similar.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR.Show that triangle ABC and triangle PQR are similar.?.
Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR.Show that triangle ABC and triangle PQR are similar.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR.Show that triangle ABC and triangle PQR are similar.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR.Show that triangle ABC and triangle PQR are similar.?.
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