Question

= A triangle has sides of lengths 18, 24 and 30 units. What is the length of the shortest

altitude of this triangle? Express your answer as a common fraction.

Answer 1

`\boxed{\boxed{\pink{\bf \leadsto The \ length \ of \ shortest \ altitude \ is \ 2.4√(6) \ units . }}}`

Explanation:

Here given measure of sides are 18 , 24 and 30 units. Firstly let's find the area of ∆ using Heron's Formula.

`\boxed{\red{\bf Area_(\triangle) =√(s(s-a)(s-b)(s-c))}}`

Where s is semi Perimeter . And here s will be ( 18 + 24 + 30 ) / 2 = 36 units .

`\bf \implies Area = √(s(s-a)(s-b)(s-c))`

`\bf \implies Area = √( 36 ( 36 - 18)(36-24)(36-30))`

`\bf \implies Area = √( 36 * 18 * 12)`

`\bf \implies Area = √( 6^2 * 6* 6 * 2 * 3)`

`\bf \implies Area = 6^2√(6) unit^2`

`\bf \boxed{ \implies Area_(triangle) = 36√(6) units^2}`

Also we know that ,

`\boxed{\red{\bf Area_(\triangle) = (1)/(2)* (base)* (height)}}`

Let's find the altitudes now ,

Altitude on side of 18 units :-

`\bf \implies Area = (1)/(2) * (base)(height) \\\\\bf \implies 36√(6) unit^2 = (1)/(2) * 18 * h_1 = 36√(6) u^2 \\\\\bf\implies h_1 =( 36\sqrt6 * 2 )/(18) \\\\\boxed{\bf\implies h_1 = 4\sqrt6 units }`

Altitude on side of 24 units :-

`\bf \implies Area = (1)/(2) * (base)(height) \\\\\bf \implies 36√(6) unit^2 = (1)/(2) * 24 * h_1 = 36√(6) u^2 \\\\\bf\implies h_1 =( 36\sqrt6 * 2 )/(24) \\\\\boxed{\bf\implies h_1 = 3\sqrt6 units }`

Altitude on side 30 units :-

`\bf \implies Area = (1)/(2) * (base)(height) \\\\\bf \implies 36√(6) unit^2 = (1)/(2) * 30 * h_1 = 36√(6) u^2 \\\\\bf\implies h_1 =( 36\sqrt6 * 2 )/(30) \\\\\boxed{\bf\implies h_1 = 2.4\sqrt6 units }`

Hence the lenght of shortest altitude is 2.4√6 units and its on the side of 30 units.

Answer 2

According to the Heron's formula, Area (A) of the triangle having sides a,b,c units is

A=

s(s−a)(s−b)(s−c)

where

s=

2

a+b+c

For the given triangle,

a=18 cm

b=24 cm

c=30 cm

s=

2

18+24+30

=36

A=

36(36−18)(36−24)(36−30)

A=

36×18×12×6

A=

216×216

=216 cm

2

Smallest side =18 cm

Area of the triangle =

2

1

×base×altitude=216

2

1

×18× altitude=216

Altitude =

9

216

=24 cm

Explanation:

hope this will help