Question

(4) In A4BC and APQR shown below. AB = QR BC = RP CA = PQ R С S A D B Р Q 1) Are CD and PS equal? Why? ii) What is the relation between the areas of AABC and APQR?​

Answer 1

Step-by-step explanation:

In Δ ABC and Δ PQR

AB = QR

BC = RP

CA = PQ

∴ Δ ABC ≅ Δ QRP {the corresponding sides on both sides of the equation are equal}

∴ ∠ABC = ∠QRP …(eq)1

In Δ CDB and Δ PSR,

∠DBC(∠ABC) = ∠SRP(∠QRP) …(eq)2 (from eq1)

∠CDB = ∠PSR = 90°

Thus, in these triangles 2 angles are equal

Since, sum of all angles of a triangle is 180°

∴ third angle is also equal

∴ ∠DCB = ∠RPS …(eq)3

According to property,

If one side of a triangle and angle at its ends are equal to one side of another triangle and the angles at its ends, then the third angles are also equal and the sides opposite equal angle are equal.

{Here, common side is BC and RP

∠DCB = ∠RPS (from eq3)

∠DBC = ∠SRP (from eq2)}

Applying it in Δ CDB and Δ PSR

CD = PS …(eq)4 (CD is opposite ∠CBD,PS is opposite ∠PRS and ∠CBD = ∠PRS)

ii) For a triangle,

∴ area(Δ ABC) =

And area (ΔPQR) =

Here, AB = QR (given)

And, CD = PS (from eq4)

Hence, areas of both the triangles are equal.