In an isosceles triangle, one of the outer corners = 60 degrees, the height right to the side is 17 cm. Find the length of the base.

Let’s designate this triangle ABC, AB = BC, AC – the base of the triangle. According to the condition, it is said that the external angle is 60 °, which means that the internal angle will be equal to 120 °. It can only be the angle at the top, ∠ B = 120 °. The angles at the base will be equal:
∠ С = ∠ А = (180 ° – 120 °) / 2 = 30 °.
Let’s make an additional construction. Let us drop the perpendicular AH on the continuation of the side СВ beyond the vertex B.
Consider a right-angled triangle AНС, in it:
∠ С = 30 °;
Leg AH = 17 cm.
The AН leg lies opposite an angle of 30 °, the hypotenuse AC = 2 * AН = 34 (cm).
In the ABC triangle, AC is the base.
Answer: the base of the AC is 34 cm.