Q4 of 24 Page 21

In Δ ABC and Δ PQR shown below.


AB = QR BC = RP CA = PQ


i) Are CD and PS equal? Why?


ii) What is the relation between the areas of Δ ABC and Δ PQR?

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.


More from this chapter

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2

In the figure, AP and BQ equal and parallel are lines drawn at the ends of the line AB. The point of intersection of PQ and AB is marked as M.


i) Are the sides of Δ AMP equal to the sides of Δ BMQ? Why?


ii) What is special about the position of M on AB?


iii) Draw a line 5.5 centimeters long. Using a set square, locate the midpoint of this line.

 3

In the figure, ABCDE is pentagon with all sides of the same length and all angles of the same size. The sides AB and AE extended, meet the side CD extended at P and Q.


i) Are the sides of Δ BPC equal to the sides of Δ EQD? Why?


ii) Are the side of AP and AQ of Δ APQ equal? Why?

 5

In the quadrilateral ABCD shown below, the sides AB and CD are parallel. M is the midpoint of the side BC.


The lines DM and AB extended, meet at N.


Are the areas of Δ DCM and Δ BMN equal? Why?

 5

In the quadrilateral ABCD shown below, the sides AB and CD are parallel. M is the midpoint of the side BC.


The lines DM and AB extended, meet at N.


What is the relation between the areas of the quadrilateral ABCD and the triangle AND?