The lengths of the sides of triangle ABC are in the ratio 4 : 3 : 5, and its perimeter is 144 cm. Find the height corresponding to the longest side.
Answers
Answer:
We will use the following formulas:
1) According to Heron’s formula, Area of triangle=s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−−√
, where, s
is the semi-perimeter, and a,b,c
are the three sides of the triangle.
2) Area of a triangle =12×base×height
Complete step by step solution:
We know that the lengths of the sides of a triangle are in the ratio 3:4:5
.
Hence, let the three sides of the triangle be 3x
, 4x
and 5x
respectively.
Now, it is given that the perimeter of the triangle is 144 cm.
Therefore, the sum of all the three sides of the triangle is 144 cm.
⇒3x+4x+5x=144
⇒12x=144
Dividing both sides by 12, we get,
⇒x=12
We will now substitute the value x
in 3x
, 4x
and 5x
to find the sides of the triangle.
Therefore, the sides of the triangle are:
3x=3×12=36cm
4x=4×12=48cm
5x=5×12=60cm
Now, the semi-perimeter of the triangle,s=1442=72
We will use Heron’s formula to find the area of the triangle because all the three sides of the triangle are known.
Substituting s=72
, a=36
, b=48
and c=60
in the formula Area of triangle=s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−−√
,we get,
Area of triangle=72(72−36)(72−48)(72−60)−−−−−−−−−−−−−−−−−−−−−−−−√
Subtracting the terms inside the bracket, we get
⇒
Area of triangle =72(36)(24)(12)−−−−−−−−−−−−−√
Simplifying the expression, we get
⇒
Area of triangle =288×3=864
Therefore, Area of the given triangle=864cm2
Now, as we can clearly see, the longest side of this triangle is of the length 60 cm.
Substitute the area of triangle=864cm2
, the base as 60 cm and height as h
, in the formula Area of a triangle =12×base×height
, we get
⇒864=12×60×h
⇒864=30×h
Dividing both sides by 30, we get
⇒8643×10=h
⇒h=28810
Converting this as decimal,
⇒h=28.8cm
Therefore, the area of the triangle is 864 square centimetres and the height corresponding to the longest side is 28.8 cm.
Hence, this is the required answer