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!!!!
Answers
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".