Question

A triangle has sides 35 cm ,54 cm ,and 61 cm long . Find its area . Also find the smallest of its altitude

Answer 1

Area = 420√5 and height = 24√5

Answer 2

$\Large{\textbf{\underline{\underline{According\;to\;the\;Question}}}}$

Here,

Sides = 35 cm, 54 cm & 61 cm

Assume

Sides,

a = 35 cm

b = 54 cm

c = 61 cm

Now,

Semi Perimeter of the Triangle,

s = (a + b + c)/2

s = (35 + 54 + 61)/2

s = 150/2

s = 75 cm

Now,

Using Heron formula :-

A = √s(s - a)(s - b)(s - c)

A = √75(75 - 35)(75 - 54)(75 - 61)

A = √75 × 40 × 21 × 14

A = √(25 × 3) × (10 × 4) × (3 × 7) × (2 × 7)

A = √25 × 4 × (3 × 3) × (7 × 7) × (2 × 5) × 2

A = √25 × 4 × (3 × 3) × (7 × 7) × (2 × 2) × 5

A = 5 × 2 × 3 × 7 × 2 √5

A = 420 √5

A = 420 × 2.236

A = 939.15 cm²

Now

Side 54 cm :-

Area of triangle,

A = ½ x Base x altitude

939.15 = ½ × 54 × altitude

altitude = (939.15 × 2)/54

altitude = 34.78 cm

$\rule{300}{1.5}$

Also,

Side 35 cm :-

Now,

Area of triangle,

A = ½ x Base x altitude

939.15 = ½ × 35 × altitude

altitude = (939.15 × 2)/35

altitude = 53.66 cm

$\rule{300}{1.5}$

Side 61 cm :-

Now,

Area of triangle,

A = ½ x Base x altitude

939.15 = ½ × 61 × altitude

altitude = (939.15 × 2)/61

altitude = 30.79 cm

Smallest altitude = 30.79 cm