Question

1. if a circle of radius 3 and a circle of radius 4 intersect with centers of distance 5 apart, what is the distance between the 2 intersection points?

Explanation:

Step1: Form triangle with centers and intersection

Let centers be \(O_1\), \(O_2\) (distance \(O_1O_2=5\)), intersection \(A\). Triangle \(O_1AO_2\) has sides 3,4,5 (right triangle).

Step2: Calculate height to hypotenuse

Area of triangle: \(\frac{1}{2} \times 3 \times 4 = 6\). Height \(h\) to hypotenuse: \(\frac{2 \times 6}{5} = 2.4\).

Step3: Find distance between intersections

Distance = \(2h = 2 \times 2.4 = 4.8\).

(Note: 4.8 is equivalent to \(\frac{24}{5}\) if fractional form is preferred, but 4.8 is a valid decimal answer.)

Answer:

4.8