Question

find the area of an isosceles triangle with two equal sides as five centimeter each and the third side is eight centimeter​

Answer 1

Answer:

answer for the given problem is given

Answer 2

Answer:

12cm²

Step-by-step explanation:

We can solve this in three methods,

1st method (Heron's Formula)(9th Grade above)

Let the sides be a, b and c

Thus,

a = 5cm, b = 5cm, c = 8cm

Now,

s = (a + b + c)/2 = (5 + 5 + 8)/2 = 18/2

s = 9

Now using Heron's formula

√(s(s - a)(s - b)(s - c))

√(9 × (9 - 5)(9 - 5)(9 - 8))

√(9 × 4 × 4 × 1)

√(144) = 12cm²

Thus,

Area of Triangle = 12cm²

2nd method (8th and below)

Given:- An isosceles Triangle ABC whose sides are 5cm, 5cm and 8cm

To find:- Area of Triangle

Construction:- Draw an altitude from A to side BC such that it is perpendicular and mark that point on BC as D

Proof:-

We know,

AB = 5cm, AC = 5cm and BC = 8cm

also,

BD = 4cm (Prperties of an Isosceles Triangle)

that is, The altitude drawn perpendicular to the base, bisects the base of the Triangle (Theorem)

Now, by Pythagoras Theorem

AB² = BD² + AC²

BD² = AB² - AC²

BD² = 5² - 4² = 25 - 16 = 9

BD = √9 = 3 cm

Thus,

Area of triangle = (1/2) × base × altitude

= (1/2) × 8 × 3 = 4 × 3

Area of Triangle = 12 cm²

3rd method (derivation method)

We know that,

Area of a Triangle = (1/2) × b × h

From the 2nd method,

we said that,

h² = a² - (b/2)²

h² = a² - (b²/4)

h² = (4a²/4) - (b²/4)

h² = (4a² - b²)/4

h = √((4a² - b²)/4)

h = √(4a² - b²)/√4

h = (1/2) × √(4a² - b²)

so, area of Isosceles triangle

= (1/2) × b × ( (1/2) × √(4a² - b²) )

= (1/2) × b × (1/2) × √(4a² - b²)

Thus,

Area of Isosceles Triangle = b/4 × √(4a² - b²)

where a is the equal sides and b is the 3rd side

we know that a = 5cm and b = 8cm

Area = (8/4) × √(4(5)² - 8²)

= 2 × √(4 × 25 - 64)

= 2 × √(100 - 64)

= 2 × √(36)

= 2 × 6

Thus,

Area of Triangle = 12cm²

Hope you understood it........All the best