Now let's recall Theorem 10.16, it will help us to evaluate r.
Theorem 10.16
If two secants intersect in the exterior of a circle, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment.
In our diagram, the products of the secants and their outer segments are equal. With this, we can write and solve an equation in terms of r.
Note that we end with a quadratic equation. To solve this equation we can use the Quadratic Formula.
ar^2+ br+ c=0 ⇔ r=- b± sqrt(b^2-4 a c)/2 a
We first need to identify the values of a, b, and c.
r^2+18.5r-36=0 ⇕ 1r^2+ 18.5r+( - 36)=0
We see that a= 1, b= 18.5, and c= - 36. Let's substitute these values into the Quadratic Formula.
Using the Quadratic Formula, we found that the solutions of the given equation are - 18.5± sqrt(486.25)2. Thus, the solutions are r_1= - 18.5+ sqrt(486.25)2≈ 1.8 and r_2= - 18.5- sqrt(486.25)2≈ - 20.3. Since the length of a segment is always a positive number, the only correct solution is r_1≈ 1.8.
Finding q
Let's now focus on the tangent and one of the secants.
This time we will start by recalling Theorem 10.17, which tells us about the segments of secants and tangents.
Theorem 10.17
If a tangent and a secant intersect in the exterior of a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant and its external secant segment.
According to the theorem, the product of the secant and its outer segment equals the square of the tangent. Let's write and solve an equation for q.
Note that we end with a quadratic equation. To solve this equation we will again use the Quadratic Formula.
First let's identify the values of a, b, and c in our equation.
q^2+16q-225=0 ⇕ 1q^2+ 16q+( - 225)=0
We see that a= 1, b= 16, and c= - 225. Let's substitute these values into the formula.
Using the Quadratic Formula, we found that the solutions of the given equation are - 8 ± 17. Thus, the solutions are q_1=- 8 +17=9 and q_2=- 8-17=- 25. Since the length of a segment is always a positive number, the only correct solution is q_1=9.
