Case Study Ankita took part in a Rangoli Competition. She made a beautiful Rangoli in the shape of a triangle ABC as shown. In the triangle, P, Q and R are mid-points of the sides AB, BC and CA respectively. She decorated it by putting a garland along the sides of △PQR. The lengths of the sides of the triangle are AB = 20 cm, BC = 26 cm and AC = 24 cm. Based on this information, answer the following questions: 1. The length of AP is : (a) QB (b) QR (c) AR (d) RP 2. The length of PQ is : (a) 12 cm (b) 13 cm (c) 10 cm (d) 15 cm 3. The length of the garland is : (a) 30 cm (b) 32 cm (c) 35 cm (d) 40 cm 4. Area of △PQR is : (a) area of △ABC(b) area of △ABC(c) area of △ABC(d) area of △ABC 5. APQR is a : (a) Rectangle (b) Parallelogram (c) Square (d) Rhombus

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KnowledgeBoat · Accepted Answer

1\. Given, P is the mid-point of AB. ⇒ AP = AB ..........(1) By mid-point theorem, The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it. Since, Q and R are the mid-points of BC and AC respectively. ⇒ QR = AB ........(2) From equation (1) and (2), we get : ⇒ AP = QR Hence, option (b) is the correct option. 2\. By mid-point theorem, The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it. Since, P and Q are the mid-points of AB and BC respectively. PQ || AC ⇒ PQ = × 24 = 12 cm. Hence, option (a) is the correct option. 3\. By mid-point theorem, The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it. Since, P and Q are the mid-points of AB and BC respectively. PQ || AC ⇒ PQ = × 24 = 12 cm. Since, P and R are the mid-points of AB and AC respectively. PR || BC ⇒ PR = × 26 = 13 cm. Since, R and Q are the mid-points of AC and BC respectively. QR || AB ⇒ QR = × 20 = 10 cm. Length of the garland = PQ + QR + PR = 12 + 10 + 13 = 35 cm. Hence, option (c) is the correct option. 4\. By mid-point theorem, The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it. In △ARP and △PQR, Since, R and Q are the mid-points of AC and BC respectively. QR || AB ⇒ QR || AP Since, P and Q are the mid-points of AB and BC respectively. PQ || AC ⇒ PQ || AR ⇒ ∠ARP = ∠QPR (Alternate angles are equal) ⇒ ∠APR = ∠QRP (Alternate angles are equal) ⇒ PR = PR (Common side) ∴ △ARP ≅ △PQR (By A.S.A axiom) ...(1) In △QRC and △PQR, ⇒ ∠CRQ = ∠PQR (Alternate angles) ⇒ ∠CQR = ∠PRQ (Alternate angles) ⇒ QR = QR (Common side) ∴ △QRC ≅ △PQR (By A.S.A axiom) ...(2) In △PBQ and △PQR, Since, R and Q are the mid-points of AC and BC respectively. QR || AB ⇒ QR || AP Since, P and Q are the mid-points of AB and BC respectively. PQ || AC ⇒ PQ || AR ⇒ ∠BPQ = ∠PQR (Alternate angles) ⇒ ∠BQP = ∠QPR (Alternate angles) ⇒ PQ = PQ (Common side) ∴ △PBQ ≅ △PQR (By A.S.A axiom) ...(3) From eq.(1), (2) and (3), we have: Area of △PBQ = Area of △QRC = Area of △ARP = Area of △PQR From figure, ⇒ Area of △ABC = Area of △PBQ + Area of △QRC + Area of △ARP + Area of △PQR ⇒ Area of △ABC = Area of △PQR + Area of △PQR + Area of △PQR + Area of △PQR ⇒ Area of △ABC = 4 Area of △PQR ⇒ Area of △PQR = Area of △ABC Hence, option (d) is the correct option. 5\. By mid-point theorem, The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it. In △ARP and △PQR, Since, R and Q are the mid-points of AC and BC respectively. QR || AB ⇒ QR || AP ⇒ QR = ⇒ QR = AP (∵ P is the midpoint of AB) Since, P and Q are the mid-points of AB and BC respectively. PQ || AC ⇒ PQ || AR ⇒ PQ = ⇒ PQ = AR (∵ R is the midpoint of AC) Since, in quadrilateral APQR opposite sides are parallel and equal. ∴ APQR is a parallelogram. Hence, option (b) is the correct option.

Mathematics

Mid-point Theorem

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Mathematics

Mid-point Theorem

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Assertion (A): The mid-points of the sides of a quadrilateral ABCD are joined in order to get quadrilateral PQRS. PQRS is a rhombus.Reason (R): Adjacent sides of a rhombus are equal and perpendicular to each other.A is true, R is falseA is false, R is trueBoth A and R are trueBoth A and R are false