Find the length of a tangent from a point Q at a distance of 25 cm from the center, given that the radius of the circle is 7 cm (or 24 cm tangent length with 25 cm distance).
Get the complete, step-by-step math solution for: "Find the length of a tangent from a point Q at a distance of 25 cm from the center, given that the radius of the circle is 7 cm (or 24 cm tangent leng...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Understand the Geometry
We are given a circle with center O and a point Q outside the circle. A tangent is drawn from Q to the circle, touching it at point T. The radius of the circle is OT=7 cm, and the distance from the center O to point Q is OQ=25 cm. We need to find the length of the tangent $QT.
Step 2: Apply Tangent-Radius Theorem
According to the tangent-radius theorem, the radius drawn to the point of tangency is perpendicular to the tangent. Therefore, the angle ∠OTQ is 90∘, making △OTQ a right-angled triangle.
Step 3: Use Pythagorean Theorem
Since △OTQ is a right-angled triangle with the right angle at T, we can apply the Pythagorean theorem. The hypotenuse is OQ, and the other two sides are OT and $QT.
Step 4: Substitute Values and Solve for QT
Substitute the given values OQ=25 cm and OT=7 cm into the Pythagorean theorem. Then, solve the equation for QT by isolating QT2 and taking the square root.