Answer:
√190
Step-by-step explanation:
In the figure , there are 2 right angled triangles with a common perpendicular & both the triangles combine to form a new right angled triangle.
Let the triangle with 9 as base be T¹ & Let the triangle with base 10 be T². Let the triangle formed by T¹ & T² be T³.
In T² ,
Hypotenuse = y
Base = 10
According to Pythagorean Theorem ,
(Hypotenuse)² = (Base)² + (Perpendicular)²
Hence, (Perpendicular)² = [tex]y^2 - 10^2 = y^2 - 100[/tex]
In T¹ ,
Perpendicular = [tex]\sqrt{y^2 - 100}[/tex] (∵ Both T¹ & T² have common perpendicular)
⇒(Perpendicular)² = [tex]y^2 - 100[/tex]
Base = 9
⇒ (Base)² = 9²
Hypotenuse =
Using Pythagorean Theorem ,
(Hypotenuse)² = (Perpendicular)² + (Base)²
⇒ (Hypotenuse)² = [tex]y^2 - 100 + 9^2[/tex] .............................................eqn.2
Now in T³ ,
Base = y
⇒ (Base)² = y²
Perpendicular = [tex]\sqrt{(y^2 - 100) + 9^2}[/tex] (∵Perpendicular of T³ = Hypotenuse of T²)
⇒ (Perpendicular)² = [tex](\sqrt{(y^2 - 100) + 9^2})^2= (y^2 - 100) + 81 = y^2 - 19[/tex]
Hypotenuse = 9 + 10 = 19
Using Pythagorean Theorem ,
(Hypotenuse)² = (Perpendicular)² + (Base)²
[tex]=> 19^2 = y^2 - 19 + y^2\\\\=> 2y^2 = 19^2 + 19 = 19(19 + 1) = 19*20\\\\=> y^2 = \frac{19*20}{2} = 19*10 = 190\\ \\=> y =\sqrt{190}[/tex]
