### Geometry Problem: Tangent Line to a Circle#### Problem Statement:Line BC is a tangent to circle A at point C. Given:- \( B \) to \( C \) is 24 inches- \( D \) to \( C \) is 16 inchesFind the length of segment \( A \) to \( C \).#### Diagram Description:- The diagram shows a circle with center \( A \).- Line \( BC \) is tangent to the circle at point \( C \).- Point \( D \) lies on the circle, perpendicular to line \( BC \), making \( \overline{AD} \) a radius of the circle.- The length \( \overline{BC} = 24 \) inches.- The length \( \overline{DC} = 16 \) inches.#### Solution Approach:To find the length of \( \overline{AC} \), we can use the Pythagorean theorem in the right triangle \( \Delta ADC \) where:- \( \overline{AD} \) is the radius of the circle and is perpendicular to the tangent \( \overline{BC} \).- \( \overline{DC} \) is the distance from point \( D \) (a point on the circle intersecting the radius) to the point of tangency \( C \).Given:- \( \overline{DC} = 16 \) inches (part of the radius perpendicular to the tangent)- \( \overline{BC} = 24 \) inches (the entire tangent segment)Since \( D \) is directly above \( B \) on the circle:- The radius \( \overline{AC} \) can be calculated using Pythagoras' theorem in \( \Delta ADC \).Using Pythagoras’ theorem:\[ \overline{AD}^2 = \overline{AC}^2 - \overline{DC}^2 \]\[ \overline{AD} = \sqrt{\overline{AC}^2 - \overline{DC}^2} \]Plugging in the given values:\[ \overline{AD} = \sqrt{24^2 - 16^2} \]\[ \overline{AD} = \sqrt{576 - 256} \]\[ \overline{AD} = \sqrt{

Name: ### Geometry Problem: Tangent Line to a Circle#### Problem Statement:
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: ### Geometry Problem: Tangent Line to a Circle#### Problem Statement:Line BC is a tangent to circle A at point C. Given:- \( B \) to \( C \) is 24 inches- \( D \) to \( C \) is 16 inchesFind the length of segment \( A \) to \( C \).#### Diagram Desc