P, Q and R are the mid-points of the sides BC, CA and AB respectively of a triangle ABC. PR and BQ meet at X, CR and PQ meet at Y and BC = 8 cm. Calculate the length of XY.
Mid-point Theorem
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Question

P, Q and R are the mid-points of the sides BC, CA and AB respectively of a triangle ABC. PR and BQ meet at X, CR and PQ meet at Y and BC = 8 cm. Calculate the length of XY.

KnowledgeBoat · Accepted Answer

2 cm Reason By mid-point theorem, The line joining mid-point of any two sides of a triangle is parallel and is equal to half of third side. Since, P and Q are mid-points of sides BC and AC, ∴ PQ || AB and PQ = AB ⇒ PQ || BR and PQ = BR [As, R is mid-point of AB, so BR is equal to half of AB]..................(1) Since, Q and R are mid-points of sides AC and AB, ⇒ QR || BC and QR = BC ⇒ QR || BP and QR = BP [As, P is mid-point of BC, so BP is equal to half of BC]..................(2) Since, P and R are mid-points of sides BC and AB, PR || AC and PR = AC From equation (1) and (2), we get : PQ || BR, QR || BP, PQ = BR, QR = BP. Since, opposite sides of the quadrilateral PQRB, are equal and parallel. ∴ PQRB is a || gm. From figure, ⇒ PR = and PR || AC [By mid-point theorem] ⇒ PR || QC and PR = QC [As, Q is mid-point of AC, so QC is equal to half of AC]..................(3) Also, ⇒ RQ = and RQ || BC [By mid-point theorem] ⇒ RQ = PC [P is mid-point of BC] and RQ || PC We know that, In parallelogram, diagonals bisects each other. PQRB is a parallelogram. Diagonal PR and QB meet at X. So, we can also say that X is mid point of PR and QB. PCQR is a parallelogram. Diagonal PQ and CR meet at Y. So, we can also say that Y is mid point of PQ and CR. In ΔRPQ, X is mid-point of RP and Y is mid-point of PQ, by mid-point theorem : ∴ XY = ⇒ XY = = = 2 cm.

Mathematics

P, Q and R are the mid-points of the sides BC, CA and AB respectively of a triangle ABC. PR and BQ meet at X, CR and PQ meet at Y and BC = 8 cm. Calculate the length of XY.

Mid-point Theorem

Related Questions

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :
(i) AD is perpendicular to FE.
(ii) AD and FE bisect each other.

Mathematics

P, Q and R are the mid-points of the sides BC, CA and AB respectively of a triangle ABC. PR and BQ meet at X, CR and PQ meet at Y and BC = 8 cm. Calculate the length of XY.

Mid-point Theorem

Related Questions

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :(i) AD is perpendicular to FE.(ii) AD and FE bisect each other.

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :
(i) AD is perpendicular to FE.
(ii) AD and FE bisect each other.