In an isosceles triangle, the sides are respectively equal to 12,10,10. Find the area.

1.) First, we write down the formula by which we will further find the area of ​​the triangle.
S = 1 / 2ah, where a is the base of the triangle and h is its height.
2.) Recall the properties of an isosceles triangle.
The two sides of such a triangle are equal, and its height is also the median.
Now let us denote the considered triangle ABC, where BD is its height and AC is its base.
Moreover, AB = BC.
Now consider a rectangular ΔABD.
Let’s find the size of the side AD.
AD = AC: 2.
AD = 12: 2.
AD = 6.
3.) Now we find out what the height of this triangle is equal to.
To do this, consider a rectangular ΔABD.
In order to find the height BD, we use the Pythagorean theorem.
According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of two legs.
That is, c² = a² + b², where a and b are the legs of the triangle, and c is its hypotenuse.
Now let’s write down for ΔABC and find the second leg of the triangle.
AB² = AD² + BD².
10² = 6² + BD².
BD² = 10² – 6².
BD² = 100 – 36.
BD² = 64.
BD² = 8².
BD = 8.
4.) Next, we calculate the area ΔABC.
S = 1 / 2ah.
For ΔABC, a is the base.
S = 1/2 × 12 × 8.
S = 1/2 × 96.
S = 48.
Answer: 48.