Base D is the height CD of triangle ABC, lies on the side AB = BC. Find AC if AB = 3, CD = √3.

Let’s use the picture for the solution.

By the condition AD = BC, and CD = √3 and is the height, AB = 3. Then we have two right-angled triangles CDB and ACD.

Consider a right-angled triangle CDB. By the Pythagorean theorem BC² = CD² + BD², then BD² = BC² – CD².

BD² = ВС² – √3 ².

BC = AD = AB – BD = 3 – BD.

BD² = (3 – BD) ² – √3 ².

BD² = (9 – 2 x 3 x BD + BD – 3 = 6 – 6 x BD + BD².

Bringing similar terms, we get:

6 x BD = 6.

BD = 1.

Then AD = AB – BD = 3 – 1 = 2.

Consider a right-angled triangle ACD and find the hypotenuse AC.

AC² = AD² + CD² = 2² + √3² = 4 + 3 = 7.

AC = √7.

Answer: AC = √7.