Answer
Verified
425.4k+ views
Hint: Here the given triangle is a right angled triangle, we can use Pythagoras theorem.
In the given figure, $\Delta GMP$ is a right angled triangle. So using Pythagoras theorem, we’ll get:
$
{(MP)^2} + {c^2} = {a^2}, \\
\Rightarrow {(MP)^2} = {a^2} - {c^2} .....(i) \\
$
$\Delta MNP$ is also a right angled triangle. So using Pythagoras theorem again, we’ll get:
$
{(MP)^2} + {d^2} = {b^2}, \\
\Rightarrow {(MP)^2} = {b^2} - {d^2} .....(ii) \\
$
From equation $(i)$ and $(ii)$, we have:
$
\Rightarrow {a^2} - {c^2} = {b^2} - {d^2}, \\
\Rightarrow {a^2} - {b^2} = {c^2} - {d^2}, \\
\Rightarrow (a + b)(a - b) = (c + d)(c - d), \\
\Rightarrow \dfrac{{(a - b)}}{{(c - d)}} = \dfrac{{(c + d)}}{{(a + b)}}. \\
$
This is the required proof.
Note: In any triangle, perpendicular drawn from a vertex to the opposite side always divides the triangle into two right angled triangles. We can use Pythagoras theorem for both the triangles separately.
In the given figure, $\Delta GMP$ is a right angled triangle. So using Pythagoras theorem, we’ll get:
$
{(MP)^2} + {c^2} = {a^2}, \\
\Rightarrow {(MP)^2} = {a^2} - {c^2} .....(i) \\
$
$\Delta MNP$ is also a right angled triangle. So using Pythagoras theorem again, we’ll get:
$
{(MP)^2} + {d^2} = {b^2}, \\
\Rightarrow {(MP)^2} = {b^2} - {d^2} .....(ii) \\
$
From equation $(i)$ and $(ii)$, we have:
$
\Rightarrow {a^2} - {c^2} = {b^2} - {d^2}, \\
\Rightarrow {a^2} - {b^2} = {c^2} - {d^2}, \\
\Rightarrow (a + b)(a - b) = (c + d)(c - d), \\
\Rightarrow \dfrac{{(a - b)}}{{(c - d)}} = \dfrac{{(c + d)}}{{(a + b)}}. \\
$
This is the required proof.
Note: In any triangle, perpendicular drawn from a vertex to the opposite side always divides the triangle into two right angled triangles. We can use Pythagoras theorem for both the triangles separately.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Two charges are placed at a certain distance apart class 12 physics CBSE
Difference Between Plant Cell and Animal Cell
What organs are located on the left side of your body class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
The planet nearest to earth is A Mercury B Venus C class 6 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is BLO What is the full form of BLO class 8 social science CBSE