Question

In the given figure, DEFG is a square and angle \[\angle BAC{\text{ }} = {\text{ }}90^\circ \].Show that \[D{E^2} = BD \times EC\].

Answer 1

Hint:First we will prove \[\Delta DBG \sim \Delta AGF\]and \[\Delta EFC \sim \Delta AGF\] using properties of square and parallel lines and criterions of similarity. Then we will prove \[\Delta EFC \sim \Delta DBG\] and equate the ratios of sides of the triangles to get the answer.

Complete step-by-step answer:
Since the opposite sides of a square are parallel
Therefore, \[DE||FG\]
Now AB being a transversal,
\[\angle GBD = \angle AGF\](corresponding angles)……………………………(1)
Similarly AC being a transversal,
\[\angle FCE = \angle AFG\](corresponding angles)……………………………(2)
In \[\Delta DBG\] and \[\Delta AGF\]
\[\angle GBD = \angle AGF\](corresponding angles) (from (1))
\[\angle GDB = \angle GAF\left( {{{90}^ \circ }} \right)\]
Therefore by AA criterion \[\Delta DBG \sim \Delta AGF\]……………………(3)
In \[\Delta EFC\] and \[\Delta AGF\]
\[\angle FCE = \angle AFG\](corresponding angles) (from (2))
\[\angle FEC = \angle GAF\left( {{{90}^ \circ }} \right)\]
Therefore by AA criterion \[\Delta EFC \sim \Delta AGF\]…………………….(4)
Now since from (3) and (4)
\[\Delta DBG \sim \Delta AGF\]and \[\Delta EFC \sim \Delta AGF\]
Therefore,
\[\Delta EFC \sim \Delta DBG\]
Now since the ratio of corresponding sides of similar triangles is equal therefore,
\[\dfrac{{CE}}{{GD}} = \dfrac{{FE}}{{BD}}..........(5)\]
Now since DEFG is square and all the sides of a square are equal
So, \[DE = EF = FG = GD\]
Therefore substituting the values in equation 5 we get:-
\[
  \dfrac{{CE}}{{DE}} = \dfrac{{DE}}{{BD}} \\
By cross multiplying we get
  D{E^2} = CE \times BD \\
 \]
Hence proved.

Note:Two triangles can be similar by following criterions:
SSS criterion (side,side,side):- In this criterion, the corresponding sides of the two triangles are in the same ratio.
AA criterion (Angle, angle) - In this criterion, the corresponding angles of the two triangles are equal.
SAS criterion(Side angle side)- In this criterion, the two sides of the triangles are in the same ratio and the corresponding angle between them is equal.
ASA criterion ( Angle, side angle)- In this criterion, two corresponding angles of the triangles are equal and the side containing them is in the same ratio.