Q13 of 49 Page 26

In figure 1.81, the vertices of square DEFG are on the sides of ΔABC, ∠A = 90°. Then prove that DE2 = BD × EC

(Hint : Show that ΔGBD is similar to ΔDFE. Use GD = FE = DE.)


Proof:In □ DEFG is a square


⇒ GF||DE


⇒ GF||BC


Now,In Δ AGF and Δ DBG


⇒ ∠ AGF≅ ∠ DBG (corresponding angles)


⇒ ∠ GDB≅ ∠ FAG (Both are 90° )


⇒ Δ AGF~Δ DBG ……(1) (AA similarity)


Now,In Δ AGF and Δ EFC


⇒ ∠ AFG≅ ∠ ECF (corresponding angles)


⇒ ∠ GAF≅ ∠ FEC (Both are 90° )


⇒ Δ AGF~Δ EFC ……(2) (AA similarity)


From (1) & (2), we have-


⇒ Δ EFC~Δ DBG


⇒


⇒ EF× DG = BD× EC


Now,∵ DEFG is a square


⇒ DE = EF = DG


⇒ DE× DE = BD× EC


⇒ DE2 = BD× EC


More from this chapter

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9

In ΔPQR seg PM is a median. Angle bisectors of ∠PMQ and ∠PMR intersect side PQ and side PR in points X and Y respectively. Prove that XY || QR.


Complete the proof by filling in the boxes. In ΔPMQ, ray MX is bisector of ∠PMQ.


.......... (I) theorem of angle bisector.


In ΔPMR, ray MY is bisector of ∠PMR.


.......... (II) theorem of angle bisector.


But .......... M is the midpoint QR, hence MQ = MR.



∴ XY || QR .......... converse of basic proportionality theorem.

 10

In fig 1.78, bisectors of ∠B and ∠C of ΔABC intersect each other in point X. Line AX intersects side BC in point Y. AB = 5, AC = 4, BC = 6 then find

11

In □ABCD, seg AD || seg BC. Diagonal AC and diagonal BD intersect each other in point P. Then show that

12

In fig 1.80, XY || seg AC. If 2AX = 3BX and XY = 9. Complete the activity to find the value of AC.


Activity :


.......... by componendo.


.......... (I)


.......... test of similarity.


.......... corresponding sides of similar triangles.


...from (I)