Here Q and S are the points on smaller and larger circles respectively and P is the center of both circles.
Given that, radius of smaller circle, PQ= 2.5
and radius of larger circle, PS= 5
RS is a chord of the larger circle and tangent to the smaller circle at point Q.
That's why, ∠PQS will be 90° and the ΔPQS is a right angle triangle.
For ΔPQS, we will use Pythagorean Theorem, according to this theorem here:
(PS)² = (PQ)² + (QS)²
⇒ (5)² = (2.5)² + (QS)² [By plugging PS= 5 and PQ= 2.5]
⇒ 25 = 6.25 + (QS)²
⇒ 18.75 = (QS)² [Subtracting 6.25 from both sides]
⇒ QS = √18.75
⇒ QS = 4.3301
Here Q is the midpoint of the cord RS, so RQ= QS
That means, RS = 2× QS
= 2× 4.3301
= 8.6602
= 8.7 [Rounding to the nearest tenth]
So, the length of RS to the nearest tenth is 8.7