Final answer:
In an isosceles triangle with base angles of 72° and an angle at the vertex of 36°, the lengths of the legs AB and BC are approximately 0.9511 units long.
Step-by-step explanation:
In an isosceles triangle, the base angles are equal. Therefore, the angles at A and C each measure 72°. The remaining angle at B measures 36°.
Since the base of the triangle is 1 unit, the length of the legs AB and BC can be determined using trigonometric functions. In this case, we can use the sine function.
The sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse. In this triangle, the hypotenuse is the base of 1 unit. Therefore, AB and BC are equal to sin(72°) times 1 unit.
Using a scientific calculator or software, we can determine that sin(72°) ≈ 0.9511. Therefore, AB and BC are approximately 0.9511 units long.