Question

the length of hypotenuse of a right angled triangle is 1 unit less that twice the length of its shortest side and the other side is 1 unit more than the shortest side. obtain a mathematical expression to find the length of shortest side of the triangle

please help me!!!!

Answer 1

Step-by-step explanation:

Given :-

The length of hypotenuse of a right angled triangle is 1 unit less that twice the length of its shortest side and the other side is 1 unit more than the shortest side.

To find :-

Obtain a mathematical expression to find the length of shortest side of the triangle ?

Solution :-

Let the shortest side of a right angled triangle be X units

Twice of the shortest side = 2X units

Length of the hypotenuse of the right angled triangle

= 1 unit less that twice the length of its shortest side

= (2X-1) units

Length of the other side of the right angled triangle

= 1 unit more than the shortest side

= (X+1) units

Given triangle is a right angled triangle,So We can apply Pythagoras Theorem

Pythagoras Theorem: "In a right angled triangle The square of the hypotenuse is equal to the sum of the squares of the other two sides".

Hypotenuse² = Other side²+Shortest side²

=> (2X-1)² = (X+1)²+X²

=> (2X)²-2(2X)(1)+1² = X²+2(X)(1)+1²+X²

Since (a+b)² = a²+2ab+b²

(a-b)² = a²-2ab+b²

=> 4X²-4X+1 = X²+2X+1+X²

=> 4X²-4X+1 = 2X²+2X+1

=> 4X²-4X+1-2X²-2X-1 = 0

=> (4X²-2X²)+(-4X-2X)+(1-1) = 0

=> 2X²+(-6X)+(0) = 0

=> 2X²-6X = 0

=> 2(X²-3X) = 0

=> X²-3X = 0/2

=> X²-3X = 0

This is the required equation is X²-3X = 0

Answer:-

X²-3X = 0 is the required equation to obtain to find the length of shortest side of the given right angled triangle.

Check :-

The equation is X²-3X = 0

=> X(X-3) = 0

=> X= 0 or X-3 = 0

=> X= 0 or X = 3

X can not be negative .

So, X = 3 units

Shortest side = 3 units

Hypotenuse = 2X-1 = 2(3)-1 = 6-1 = 5 units

Other side = X+1 = 3+1 = 4 units

The sides of the given right angled triangle are

3 units , 4 units and 5 units

Used formulae:-

• (a+b)² = a²+2ab+b²
• (a-b)² = a²-2ab+b²

Pythagoras Theorem:

• "In a right angled triangle The square of the hypotenuse is equal to the sum of the squares of the other two sides".