CLASS 10TH MATHS CH-6 THEOREM 6.8 AND THEOREM 6.7

Theorem 6.8 states that, In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

We have to prove this theorem. 

Before proving we need to know theorem 6.7. Because we will use theorem 6.7 for proving theorem 6.8.

Now theorem 6.7 states that, If a perpendicular is drawn from the vertex of the right angle triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

It means that in a right triangle ABC right angled at B .when a perpendicular is drawn from angle B to AC then the two triangles formed namely as triangle ABD and triangle BDC are similar to each other and to whole triangle ABC.

SO, NOW WE KNOW THE THEOREM 6.7 AND WITH THE HELP OF THIS THEOREM WE WILL PROVE THE THEOREM 6.8.

To prove theorem 6.8, we are given a right triangle ABC right angled at B.

To prove: AC² = AB² + BC²

Construction: we have to construct a perpendicular BD from Angle B to line AC.

Proof: we know that 

         Triangle ADB~ Triangle ABC ( from theorem 6.7 )

         

In similar triangles the ratio of sides is equal .

           So,  AD = AB

                  AB     AC

               AD × AC = AB × AB

               AD × AC = AB² (1)

              

Also,Triangle BDC~ Triangle ABC (from theorem 6.7)

        So,  CD = BC  ( sides are proportional )

               BC     AC

            CD × AC = BC × BC

            CD × AC = BC² (2)

By adding (1) and (2),

          AD × AC + CD × AC = AB² + BC²

          AC( AD + CD) = AB² + BC² 

          AC × AC = AB² + BC²  (AD + CD = AC)

          AC² = AB² + BC²

To know about theorem 6.6 then click below 

THEOREM 6.6

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