Question

2. The sides of a right-angled triangle containing the
right angle are 5x and (3x - 1) cm. If the area of the
triangle be 60 cm", calculate the lengths of the sides
of the triangle
(ICSE)​

Answer 1

Answer:

Required sides of this triangle are of lengths 8 cm and 15 cm.

Step-by-step explanation:

From the properties of right angled triangles :

• Area of ∆ = 1 / 2 x height x base
• Or, 1 / 2 x product of sides { not hypotenuse }

Here,

Sides of the triangle are 5x cm and ( 3x - 1 ) cm { cm is used for unit, centimeter }.

Given,

= > Area of this triangle = 60 cm^2

= > 1 / 2 × 5x cm × ( 3x - 1 ) cm = 60 cm^2

= > 5x( 3x - 1 ) = 120

= > x( 3x - 1 ) = 24

= > 3x^2 - x = 24

= > 3x^2 - x - 24 = 0

= > 3x^2 - ( 9 - 8 )x - 24 = 0

= > 3x^2 - 9x + 8x^2 - 24 = 0

= > 3x( x - 3 ) + 8( x - 3 ) = 0

= > ( x - 3 )( 3x + 8 ) = 0

Case 1 : If x - 3 is 0

= > x - 3 = 0

= > x = 3

Case 2 : If 3x + 8 is 0

= > 3x + 8 = 0

= > 3x = - 8

= > x = - 8 / 3

= > x = 3 or - 8 / 3.

Since length can't be negative, value of x is 3.

Hence length of sides :

• 5x cm = 5( 3 ) cm = 15 cm
• 3x - 1 cm = 3( 3 ) - 1 cm = 8 cm.

Hence the required sides of this triangle are of lengths 8 cm and 15 cm.

Answer 2

Answer:

${\pink{\green{\therefore Sides\:are=15cm\:and\:8cm}}}$

Step-by-step explanation:

$\huge{\pink{\green{\underline{\red{\sf{SOLUTION-}}}}}}$

▪In the given question information given about relation between two Sides of a Right Angled Triangle and Area of this Triangle is given.

▪We have to find Sides of this Triangle.$\underline \bold{Given : } \\ \implies Base =( 3x - 1) \\ \implies Height = 5x \\ \implies Area \: of \: Triangle =60 {cm}^{2} \\ \\ \underline \bold {To \: Find : } \\ \implies Length \: of \: Sides \: are = ?$

▪According to given question.

▪We know the formula two find Area of Triangle that is half multiply Base multiply Height.

$\implies Area \: of \: Triangle = 60 {cm}^{2} \\ \implies \frac{1}{2} \times Base \times Height = 60 {cm}^{2} \\ \implies \frac{1}{2} \times (3x - 1)(5x) = 60 \\ \implies 15 {x}^{2} - 5x = 120 \\ \implies 5( {3 x }^{2} - x) = 120 \\ \implies 3 {x}^{2} - x = 24 \\ \implies 3{x}^{2} - x - 24 = 0 \\ \implies {3x}^{2} - 9x + 8x - 24 = 0 \\ \implies 3x(x - 3) + 8(x - 3) = 0 \\ \implies (3x + 8)(x - 3) = 0 \\ \\ \implies 3x + 8 = 0 \\ \implies x = \frac{ - 8}{3} \\ \\ \implies (x - 3) = 0 \\ \bold{\implies x = 3} \\ \\ > > From \: these \: two \: value \: of \: x \: we \: take \\ positive \: because \: Length \: of \: triangle \\ cannot \: be \: negative. \\ \\ \bold{ \implies Height = 5x = 5 \times 3 = 15 \: cm} \\ \bold{ \implies Base = 3x - 1 = 9 - 1 = 8 \: cm}$_________________________________________