The side of an isosceles triangle is 25 and the base is exactly 30. Find the area of this triangle.

The area of ​​a triangle is half the product of the base and the height. S = ah / 2. We know the base 30. We need to find the height drawn to this base.

In an isosceles triangle, the median is the height and bisector. This means that the height divides the base in half (since it is the median, and the median is the segment connecting the vertex with the middle of the opposite side). The height also divides the triangle into two right-angled triangles, with the hypotenuse equal to the side of the triangle, one leg is the height of the triangle, the second leg is half the base of the triangle. Let’s find the height of the triangle by the Pythagorean theorem: The square of the hypotenuse is equal to the sum of the squares of the legs.

Let’s introduce the designations: h – height (first leg), x – lateral side (hypotenuse), y – half base (second leg).

x ^ 2 = h ^ 2 + y ^ 2;

x = 25, y = 30/2 = 15;

h ^ 2 = x ^ 2 – y ^ 2;

h ^ 2 = 25 ^ 2 – 15 ^ 2 = 625 – 225 = 400;

h = √400 = 20.

S = (30 * 20) / 2 = 600/2 = 300.

Answer. 300.