In the given triangle P, Q and R are mid-points of sides AB, BC and AC respectively. Prove that triangle PQR is similar to triangle ABC.
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In the given triangle P, Q and R are mid-points of sides AB, BC and AC respectively. Prove that triangle PQR is similar to triangle ABC.

KnowledgeBoat · Accepted Answer

In ∆ABC, Since, P is mid-point of AB and R is mid-point of AC so, By mid-point theorem, PR || BC. By basic proportionality theorem we have, And, in ∆PAR and ∆BAC, ∠PAR = ∠BAC [Common] ∠APR = ∠ABC [Corresponding angles are equal] ∴ ∆PAR ~ ∆BAC [By AA] Since, corresponding sides of similar triangles are proportional. [∵ P is the mid-point of AB] Since, P and Q are mid-points of AB and BC so, by mid-point theorem, PQ = AC Since, R and Q are mid-points of AC and BC so, by mid-point theorem, RQ = AB So, Hence, proved that ∆PQR ~ ∆ABC by SSS similarity.

Mathematics

In the given triangle P, Q and R are mid-points of sides AB, BC and AC respectively. Prove that triangle PQR is similar to triangle ABC.

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Related Questions

In the given figure, ABC is a right angled triangle with ∠BAC = 90°.
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ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that :
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Mathematics

In the given triangle P, Q and R are mid-points of sides AB, BC and AC respectively. Prove that triangle PQR is similar to triangle ABC.

Similarity

Related Questions

In the given figure, ABC is a right angled triangle with ∠BAC = 90°.(i) Prove that : △ADB ~ △CDA.(ii) If BD = 18 cm and CD = 8 cm, find AD.(iii) Find the ratio of the area of △ADB is to area of △CDA.

ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that :(i) △ADE ~ △ACB(ii) If AC = 13 cm, BC = 5 cm and AE = 4 cm. Find DE and AD.(iii) Find, area of △ADE : area of quadrilateral BCED.

In the given figure, ABC is a right angled triangle with ∠BAC = 90°.
(i) Prove that : △ADB ~ △CDA.
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(iii) Find the ratio of the area of △ADB is to area of △CDA.

ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that :
(i) △ADE ~ △ACB
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(iii) Find, area of △ADE : area of quadrilateral BCED.