Two isosceles triangles have the same base length. the legs of one of the triangles are twice as long as the legs of the other. find the lengths of the sides of the triangles of their perimeters are 23 cm and 43 cm.

An isosceles triangle is a type of triangle where 2 sides are equal.

Picture out 2 triangles with the same base length. 
On the first triangle, its legs are twice the length of the legs of the second triangle.

To put it into variables, let:
     B = the same base length of the two triangles
     A = the length of one leg the smaller triangle
    2A = the length of one leg of the bigger triangle

Given:    Perimeter of smaller triangle = 23cm
              Perimeter of bigger triangle = 43cm

Recall the formula for solving the perimeter of a triangle:
      Perimeter = A + B + C
   
where, A, B, and C are the legs of the triangle

Since the triangle involved is an isosceles triangle, therefore, we can say that
        Perimeter = 2A + B     ,   2 legs are equal ( A=C )

Substituting the given perimeter value to the formula. 
   
23cm = 2A + B                    ⇒   equation 1  (smaller triangle)
43cm = 2(2A) + B                ⇒   equation 2  (bigger triangle)

Simplifying equation 2.
                                    43cm = 4A + B
  (rearranging)             B = 43cm - 4A        ⇒ equation 3

Substituting equation 3 to equation 1:
   (equation 1)              23cm = 2A + B
                                    23cm = 2A + (43cm - 4A)
                                    23cm = -2A + 43cm
                                    2A = 43cm - 23cm
                                    2A = 20cm  ⇒   length of the leg of the bigger triangle
                                     A = 10cm   ⇒   length of the leg of the smaller triangle

To solve for the base length, just substitute the value of A to equation 3
     (equation 3)       B = 43cm - 4A
                               B = 43cm - 4(10cm)
                               B = 3 cm

Final Answer:

• For the smaller triangle, the length of the sides are 10cm, 10cm, and 3cm
• For the bigger triangle, the length of the sides are 20cm, 20cm, and 3cm