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\[Hypotenus{e^2}\; = {\rm{ }}Perpendicula{r^2}\; + {\rm{ }}Bas{e^2}\;\]

Here, it is given that length of the ladder = 25 m

That is from the diagram we have \[AC = ED = 25\]

Also given that the distance between the wall and the foot of the ladder is 7m.

That is \[CB = {\rm{ }}7\]

Here it is also given that the ladder slides 4m in the wall.

That is \[AE = 4\]

Before sliding, the situation is represented by a right triangle with hypotenuse 25, and base 7,

Using Pythagoras theorem we have,

\[Hypotenus{e^2}\; = {\rm{ }}Perpendicula{r^2}\; + {\rm{ }}Bas{e^2}\;\]

We have to find the length of the wall, which is nothing but the perpendicular in the triangle.

\[{\rm{ }}Perpendicula{r^2}\; = Hypotenus{e^2} - {\rm{ }}Bas{e^2}\;\]

On comparing with the triangle we get,

\[A{B^2} = A{C^2} - C{B^2}\]

By substituting the values known we get,

\[A{B^2} = {25^2} - {7^2}\]

Let us square the terms in the right hand side we get,

\[A{B^2} = 625 - 49\]

By solving and taking square root on both the sides, we get

\[AB = \sqrt {576} \]

\[AB = 24{\rm{ }}m\]

Hence, the vertical arm that is the height of the wall is 24 m.

After sliding, the situation is represented by a right triangle with hypotenuse 25, and vertical arm has been slide 4 m then the vertical arm is of length \[24 - 4 = 20\]m,

Using Pythagoras theorem in this triangle, we get

\[Hypotenus{e^2}\; = {\rm{ }}Perpendicular{r^2}\; + {\rm{ }}Bas{e^2}\;\]

We have to find the distance between the wall and foot of the ladder after sliding, that is we have to find the base of the triangle.

\[{\rm{ }}Bas{e^2}\;\; = Hypotenuse{e^2} - {\rm{ }}Perpendicula{r^2}\]

On comparing with the triangle we get,

\[D{B^2} = D{E^2} - E{B^2}\]

By substituting the known values we get

\[D{B^2} = {25^2} - {20^2}\]

Let us square and solve right hand side of the equation, we get,

\[D{B^2} = 625 - 400 = 225\]

By taking square root on both the sides, we get

\[DB = \sqrt {225} \]

\[DB = 15{\rm{ }}m\]

So that the distance between foot of ladder and wall is \[15{\rm{ }}m\]

To find the distance changed in the ladder foot is \[15 - 7 = 8{\rm{ }}m\]

Hence, the distance from the foot of the ladder slide is 8 m

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