The lateral side of an isosceles triangle is 5 cm, and the height lowered to the base is 4 cm.

The lateral side of an isosceles triangle is 5 cm, and the height lowered to the base is 4 cm. Find the perimeter of the triangle.

An isosceles triangle is given.
Let’s call it ABC.
Find the perimeter ABC.
Let’s write down the formula by which we will find the perimeter of the triangle ABC.
P = AB + BC + AC.
1.) Since ABC is an isosceles triangle, then AB = BC = 5 cm.
2.) Now we find what the length of the base AC is equal to.
According to the construction, a height is drawn from the top B to the base AC.
Let’s call this height BN.
Point N bisects the base AC.
That is, AC = AN + NC.
Since AN = NC, AC = 2NC.
In a triangle BNC, the angle BNC is right, so we turn to the Pythagorean theorem.
According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
In order to find the side AC, we write by the Pythagorean theorem for the triangle BNC.
BC² = BN² + NC².
Now let’s express NC².
NC² = BC² – BN².
By condition BC = 5 cm, and BN = 4 cm.
NC² = 5² -4².
NC² = 25 – 16.
NC² = 9.
NC² = 3².
NC = 3 (cm).
Now let’s find the AC side.
AC = 2NC = 2 × 3 = 6 (cm).
3.) Since P = AB + BC + AC.
Substitute and calculate the perimeter.
P = 5 + 5 + 6 = 16 (cm).
Answer: 16 cm.