Question

Construct a triangle ABC in which BC= 3.6cm, <B=600and AB+AC=4.8 cm.

38. The length of the sides of a triangle are in the ratio 3:4:5.The perimeter of a

triangle is 144 cm. Find the area of the triangle​

Answer 1

Question:

1.) Construct a triangle ABC in which BC= 3.6cm, <B=600and AB+AC=4.8 cm.  

2.) The length of the sides of a triangle are in the ratio 3:4:5.The perimeter of a  

triangle is 144 cm. Find the area of the triangle​.

1.)  Solution:

Steps of construction:

1. Draw a line segment BC of 3.6cm.

2. At the point B, draw ∠x BC  of 60o

3. With center B and radius 4.8cm, draw an arc which intersects XB at D.

4. Join DC

5. Draw the perpendicular bisector of DC which intersects DB at A.

6. Join AC

Hence  Δ  ABCis the required triangle

2.)Perimeter = 144 cm and ratio of sides = 3 : 4 : 5.

Sum of ratio terms = 3 + 4 + 5 = 12.

Let the lengths of the sides be a, b and c respectively.

Then, a=(144×312)cm=36cm,b=(144×412)cm=48cm

and c=(144×512)cm=60cm.

∴ s=12(a+b+c)=12(36+48+60)cm=72cm.

∴ (s−a)=(72−36)cm=36cm,(s−b)=(72−48)cm−24cmand(s−c)=(72−60)cm=12cm.

(i) By Heron's formula, the area of the triangle is given by

Δ=s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√

=72×36×24×12−−−−−−−−−−−−−−−√cm2=36×24×24−−−−−−−−−−√cm2

=(36×24)cm2=864cm2. Hence, the area of the given triangle is 864 cm2.

(ii) Let base = longest side = 60 cm and the corresponding height = h cm.

Then, area =(12×base×height) sq units

=(12×60×h)cm2=(30h)cm2.

∴ 30h=864⇒h=(86430)⇒height=28.8cm.

Hence, the required height is 28.8 cm.

#Hopeithelps.