Question

Show that the sides 5cm ,12cm and 13 cm makes a right angled triangle. Find the length of the altitude on the longest side.

Answer 1

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we know according to pythagorous theorem that the square of the longest side is equal to the sum of squares of other two sides.

so,

the longest side given over here in the question is: 13cm

and the other two sides are: 12cm and 5cm.

now,

H² = B² + P²
13² = 12² + 5²
169 = 144 + 25
169 = 169

So, 

yes the triangle given over here is right angled triangle as the square of the longest side is equal to the sum of squares of the other two sides.

now, 

by construction draw an altitude on the longest side.

now let a= 5, b=12 and c= 13 

and using heron's formula we get,

s= (a + b + c) / 2 { semi perimeter }
s= 5 + 12 + 13 / 2
s= 30 / 2
s= 15


now,


Area of the triangle = √[s(s-a)(s-b)(s-c)]
                                  = √[ 15 ( 15-5 ) ( 15-12 ) ( 15-13 )
                                  = √[ 15 ( 10 ) ( 3 ) ( 2 )
                                  = √[ 15 ( 60 )
                                  = √[  900 ]
                                  = √[ 30 × 30]
                                  = 30 cm²

now we also know that area of triangle = $\frac{1}{2}$ × base × altitude
                                                                  = $\frac{1}{2}$ ×  13 ×  altitude
                                                     30 × 2  = 13 ×  altitude
                                                     60 / 13 = altitude
                                                     4.6 cm = altitude


we will be taking the hypotenuse the base as the altitude is drawn from it.


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Answer 2

Answer:

we know according to pythagorous theorem that the square of the longest side is equal to the sum of squares of other two sides.

so,

the longest side given over here in the question is: 13cm

and the other two sides are: 12cm and 5cm.

now,

H² = B² + P²

13² = 12² + 5²

169 = 144 + 25

169 = 169

So,  

yes the triangle given over here is right angled triangle as the square of the longest side is equal to the sum of squares of the other two sides.

now,  

by construction draw an altitude on the longest side.

now let a= 5, b=12 and c= 13  

and using heron's formula we get,

s= (a + b + c) / 2 { semi perimeter }

s= 5 + 12 + 13 / 2

s= 30 / 2

s= 15

now,

Area of the triangle = √[s(s-a)(s-b)(s-c)]

                                 = √[ 15 ( 15-5 ) ( 15-12 ) ( 15-13 )

                                 = √[ 15 ( 10 ) ( 3 ) ( 2 )

                                 = √[ 15 ( 60 )

                                 = √[  900 ]

                                 = √[ 30 × 30]

                                 = 30 cm²

now we also know that area of triangle =  × base × altitude

                                                                 =  ×  13 ×  altitude

                                                    30 × 2  = 13 ×  altitude

                                                    60 / 13 = altitude

                                                    4.6 cm = altitude

we will be taking the hypotenuse the base as the altitude is drawn from it.