The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ. Prove that :
ar.(△ ABC) = 8 × ar.(△ QSB)

Question

The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ. Prove that : ar.(△ ABC) = 8 × ar.(△ QSB)

KnowledgeBoat · Accepted Answer

In △ ABC, R and Q are mid-points of AC and BC respectively. ∴ RQ || AB (By mid-point theorem) We know that, Area of triangles on the same base and between the same parallel lines are equal. △ PBQ and △ PAR lie on same base (AP = BP) and between same parallel lines RQ and AB. ∴ Area of △ PBQ = Area of △ APR ...............(1) P and R are mid-points of AB and AC respectively. ∴ PR || BC (By mid-point theorem) ∴ PR || BQ ∴ PBQR is a parallelogram. PQ is the diagonal. ⇒ Area of △ PBQ = Area of △ PQR (Diagonal of parallelogram divides it into two triangles of equal areas.) .........(2) From equation (1) and (2), we get : ⇒ Area of △ PBQ = Area of △ PQR = Area of △ APR ........(3) P and Q are mid-points of AB and BC respectively. ∴ PQ || AC (By mid-point theorem) ∴ PQ || RC ∴ PQCR is a parallelogram. RQ is the diagonal. ⇒ Area of △ PQR = Area of △ RQC (Diagonal of parallelogram divides it into two triangles of equal areas.) .........(4) From equation (3) and (4), we get : ⇒ Area of △ PBQ = Area of △ PQR = Area of △ APR = Area of △ RQC = x (let). From figure, ⇒ Area of △ PBQ + Area of △ PQR + Area of △ APR + Area of △ RQC = Area of △ ABC ⇒ x + x + x + x = Area of △ ABC ⇒ 4x = Area of △ ABC ⇒ x = Area of △ ABC ⇒ Area of △ PBQ = Area of △ ABC ........(5) In △ PBQ, S is the mid-point of PQ and BS is median. ⇒ Area of △ QSB = Area of △ PSB (Median divides triangle into two triangles of equal area) ⇒ Area of △ QSB = Area of △ PBQ ⇒ Area of △ PBQ = 2 Area of △ QSB ..........(6) Substituting value of area of △ PBQ from equation (6) in (5), we get : ⇒ 2 Area of △ QSB = Area of △ ABC ⇒ Area of △ QSB = Area of △ ABC. Hence, proved that area of △ QSB = area of △ ABC.

Mathematics

The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ. Prove that :
ar.(△ ABC) = 8 × ar.(△ QSB)

Theorems on Area

Related Questions

The given figure shows parallelograms ABCD and APQR. Show that these parallelograms are equal in area.

ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm²; find the area of parallelogram.

In the given figure, the diagonals AC and BD intersect at point O. If OB = OD and AB // DC, prove that :
(i) Area of (△ DOC) = Area of (△ AOB)
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(iii) ABCD is a parallelogram.

The given figure shows a parallelogram ABCD with area 324 sq.cm. P is a point in AB such that AP : PB = 1 : 2. Find the area of △ APD.

Mathematics

The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ. Prove that :ar.(△ ABC) = 8 × ar.(△ QSB)

Theorems on Area

Related Questions

The given figure shows parallelograms ABCD and APQR. Show that these parallelograms are equal in area.

ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm2; find the area of parallelogram.

In the given figure, the diagonals AC and BD intersect at point O. If OB = OD and AB // DC, prove that :(i) Area of (△ DOC) = Area of (△ AOB)(ii) Area of (△ DCB) = Area of (△ ACB)(iii) ABCD is a parallelogram.

The given figure shows a parallelogram ABCD with area 324 sq.cm. P is a point in AB such that AP : PB = 1 : 2. Find the area of △ APD.

The following figure shows a triangle ABC in which P, Q and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ. Prove that :
ar.(△ ABC) = 8 × ar.(△ QSB)

ABCD is a parallelogram in which BC is produced to E such that CE = BC and AE intersects CD at F. If ar.(△ DFB) = 30 cm²; find the area of parallelogram.

In the given figure, the diagonals AC and BD intersect at point O. If OB = OD and AB // DC, prove that :
(i) Area of (△ DOC) = Area of (△ AOB)
(ii) Area of (△ DCB) = Area of (△ ACB)
(iii) ABCD is a parallelogram.