Question

3. The sides of a right-angled triangle are
(x - 1) cm, 3x cm and (3x + 1) cm. Find :
(i) the value of x,
(ii) the lengths of its sides,
(iii) its area.

Answer 1

Longer side = Hypotenuse = (3x + 1) cm

Longer side = Hypotenuse = (3x + 1) cm Lengths of other two sides are (x − 1) cm and 3x cm.

Using pythagoras theorem,

(3x+1)^2=(x-1)^2 +3x^2

9x^2+1+6x=x^2+1-2x+9x^2

x^2-8x=0

x(x-8)=0

x=0,8

But, if x = 0, then one side = 3x = 0, which is not possible.

So, x = 8

Thus, the lengths of the sides of the triangle are (x − 1) cm = 7 cm, 3x cm = 24 cm and (3x + 1) cm = 25 cm

Area of triangle=1/2 ×7+24 =84cm^2

hope its helpful...^_^

Answer 2

ANSWER:

Given:

• Sides of a triangle = (x-1)cm, 3x cm, (3x+1)cm

To Find:

• Value of x,
• Length of all sides, and
• Area of the triangle

Solution:

$\text{We know that, in a right-angled triangle, the hypotenuse is the longest side.}\\\\\text{And, we can see that, (3x+1) is the longest side.}\\\\\text{So,}\\\\:\implies 3x+1=Hypotenuse.\\\\\text{We know that, according to Pythagoras Theorem,}\\\\:\hookrightarrow(Hypotenuse)^2=(Base)^2+(Height)^2\\\\\text{In this question, we can take any of the other sides as base and height. So,}\\\\:\implies(3x+1)^2=(3x)^2+(x-1)^2\\\\\text{We know that,}\\\\:\hookrightarrow(a\pm b)^2=a^2\pm2ab+b^2\\\\\text{So,}\\\\:\implies(3x)^2+2(3x)(1)+1^2=(3x)^2+x^2-2(x)(1)+1^2\\\\:\implies9x^2+6x+1=9x^2+x^2-2x+1\\\\:\implies9x^2+x^2-2x+1=9x^2+6x+1\\\\\text{Transposing RHS to LHS,}\\\\:\implies10x^2-2x+1-9x^2-6x-1=0\\\\\text{Rearranging the like terms,}\\\\:\implies10x^2-9x^2-2x-6x+1-1=0\\\\:\implies x^2-8x=0\\\\:\implies x(x-8)=0\\\\:\implies x=0\:\:or\:\:x=8$

$\text{But, putting x=0 in 3x will give us 0, but the side cannot be 0. So,}\\\\\bf{:\implies x=8}\\\\\\\text{The length of each side:}\\\\1):\implies x-1=8-1=7cm\\\\2):\implies3x=3(8)=24cm\\\\3):\implies3x+1=3(8)+1=24+1=25cm\\\\\text{\bf{So, the length of each side is 7cm, 24cm and 25cm respectively.}}\\\\\\\text{We know that,}\\\\:\hookrightarrow\text{Area of a right triangle}=\dfrac{1}{2}\times base\times height\\\\\text{So,}\\\\:\implies\text{Area of the triangle}=\dfrac{1}{2\!\!\!/}\times24\!\!\!\!/^{\:12}\times7\\\\:\implies\text{Area of the triangle}=12\times7\\\\:\implies\text{Area of the triangle}=84cm^2\\\\\text{\bf{The area of the triangle is 84cm ^2 }}$

Formulae Used:

• Pythagoras Theorem = (Hypotenuse)^2 = (Base)^2 + (Height)^2
• (a ± b)^2 = a^2 ± 2ab + b^2
• Area of right triangle = 1/2 × base × height