2 isosceles triangles have equal angles opposite the bases. in one of the triangles the side

2 isosceles triangles have equal angles opposite the bases. in one of the triangles the side and the height drawn to the base are 5cm and 4cm. find the perimeter of the second triangle if its lateral side is 15cm.

Since in these triangles the angles opposite to the base are equal, then these triangles will be similar, since these triangles are isosceles, and in isosceles triangles the angles at the base are equal.

First, let’s find the length of the base of the first triangle.

Since in an isosceles triangle the height is the median and hypotenuse, the triangles ΔАВН and ΔНВС are equal and are rectangular. Thus, we can find the segment AH, which is equal to the segment NS. To do this, apply the Pythagorean theorem:

AB ^ 2 = BH ^ 2 + AH ^ 2;

AH ^ 2 = AB ^ 2 – BH ^ 2;

AH ^ 2 = 5 ^ 2 – 4 ^ 2 = 25 – 16 = 9;

AH = √9 = 3 cm.

AC = AH + HC;

AC = 3 + 3 = 6 cm.

Now let’s find the sides of the second triangle. Let’s calculate the coefficient of similarity:

k = A1B1 / AB;

k = 15/5 = 3;

B1C1 = A1B1 = 15 cm.

A1C1 = AC · k;

A1C1 = 6 3 = 18 cm.

The perimeter of a triangle is the sum of the lengths of its sides:

P = A1B1 + B1C1 + A1C1;

P = 15 + 15 + 18 = 48 cm.

Answer: the perimeter of the second triangle is 48 cm.