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GMAT - Data Sufficiency - Geometry

The hypotenuse of a right triangle is 10 cm

The hypotenuse of a right triangle is 10 cm. What is the perimeter, in centimeters, of the triangle?

(1) The area of the triangle is 25 square centimeters.
(2) The 2 legs of the triangle are of equal length.

Given hypotenuse = 10cm

let other sides of the triangle be x and y,

By Pythagorus Theorem -

$y = \sqrt { { 10 }^{ 2 }-{ x }^{ 2 } } =\quad \sqrt { 100\quad -\quad { x }^{ 2 } }$

Statement 1:

The area of the triangle is 25 square centimeters

Since we know that in a right angle triangle area can be written as = 1/2 * product of two sides that are not hypotenuse

$\quad \sfrac { 1 }{ 2 } *\quad x\quad *\quad \sqrt { 100\quad -\quad { x }^{ 2 } } \\ \\ 25\quad =\quad 1/2\quad *\quad x\quad *\quad \sqrt { 100\quad -\quad { x }^{ 2 } } \\ \\ 50\quad =\quad x\quad *\quad \sqrt { 100\quad -\quad { x }^{ 2 } } \\ \\ squaring\quad both\quad sides\\ \\ 2500\quad =\quad 100{ x }^{ 2 }\quad -\quad { x }^{ 4 }\\$

we can calculate the values of x and y from the equation using a calculator which would be $\sqrt { 50 } ,\sqrt { 50 } \quad$

Perimeter would be $10 + \sqrt { 50 } ,\sqrt { 50 } \quad$

sufficient

Statement 2:

The 2 legs of the triangle are of equal length.

We can find out the lengths of 2 sides by using pythagorus theorem and get the same result in statement 1

Sufficient