A circle is drawn with PQ as its diameter. A perpendicular is drawn to PQ at C meeting the circle at A and B. If ℓ(AC) = 4 cm and ℓ(PC) = 3 cm, then find the circumference of the circle.

Step 1: Understand the problem We are given a circle with PQ as its diameter. A perpendicular is drawn to PQ at C meeting the circle at A and B. The length of AC is 4 cm and the length of PC is 3 cm. We are asked to find the circumference of the circle.

Step 2: Use the Pythagorean theorem Since AC is perpendicular to PC, triangle APC is a right triangle. We can use the Pythagorean theorem to find the length of the hypotenuse AP, which is also the radius of the circle. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. So, we have:

AP² = AC² + PC² AP² = 4² + 3² AP² = 16 + 9 AP² = 25 AP = √25 AP = 5 cm

Step 3: Find the circumference of the circle The formula for the circumference of a circle is 2πr, where r is the radius of the circle. In this case, the radius of the circle is the length of AP, which we found to be 5 cm. So, the circumference of the circle is:

Circumference = 2πr Circumference = 2π(5 cm) Circumference = 10π cm

So, the circumference of the circle is 10π cm.