Question

Area proof of Isosceles triangle​

Answer 1

Step-by-step explanation:

To find the area of an isosceles triangle using the lengths of the sides, label the lengths of each side, the base, and the height if it's provided. Then, use the equation Area = ½ base times height to find the area.

Answer 2

Altitude of an Isosceles Triangle = √(a2 − b2/4)

Thus,

Area of Isosceles Triangle Using Only Sides = ½[√(a2 − b2 /4) × b]

Here,

b = base of the isosceles triangle

h = Height of the isosceles triangle

a = length of the two equal sides

Derivation for Isosceles Triangle Area Using Heron’s Formula

The area of an isosceles triangle can be easily derived using Heron’s formula as explained below.

According to Heron’s formula,

Area = √[s(s−a)(s−b)(s−c)]

Where, s = ½(a + b + c)

Now, for an isosceles triangle,

s = ½(a + a + b)

⇒ s = ½(2a + b)

Or, s = (a + b/2)

Now,

Area = √[s(s−a)(s−b)(s−c)]

Or, Area = √[s (s−a)2 (s−b)]

⇒ Area = (s−a)2 × √[s (s−b)]

Substituing the value of “s”

⇒ Area = (a + b/2 − a)2 × √[(a + b/2) × ((a + b/2) − b)]

⇒ Area = b/2 × √[(a + b/2) × (a − b/2)]

Or, area of isosceles triangle = b/2 × √(a2 − b2/4)