Solve the given problems. Sketch an appropriate figure,...

Solve the given problems. Sketch an appropriate figure, unless the figure is given. The diameter $d$ of a pipe can be determined by noting the distance $x$ on the $V$ -gauge shown in Fig. $4.66 .$ Points $A$ and $B$ indicate where the pipe touches the gauge, and $x$ equals either $A V$ or $V B .$ Find a formula for $d$ in terms of $x$ and $\theta$ (FIGURE CAN'T COPY)

Solve the given problems. Sketch an appropriate figure, unless the figure is given.
The diameter $d$ of a pipe can be determined by noting the distance $x$ on the $V$ -gauge shown in Fig. $4.66 .$ Points $A$ and $B$ indicate where the pipe touches the gauge, and $x$ equals either $A V$ or $V B .$ Find a formula for $d$ in terms of $x$ and $\theta$
(FIGURE CAN'T COPY)

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Key Concepts

Circle Tangency

When a circle is tangent to a line, the radius drawn to the point of tangency is perpendicular to that line. This principle is crucial in problems where the circle touches straight lines, as it provides a right angle that can be used to form right triangles and apply trigonometric relationships.

Angle Bisector and Inscribed Circles

For a circle that is tangent to two intersecting lines, the center of the circle lies on the bisector of the angle formed by those lines. This fact is instrumental in relating the distances from the vertex of the angle to the points of tangency to the circle’s radius and ultimately its diameter.

Right Triangle Trigonometry

Right triangle trigonometry, such as the use of sine, cosine, and tangent ratios, is used to relate angles and side lengths in the geometric construction. In the context of a circle tangent to two lines, the trigonometric ratios in the right triangles formed help derive formulas connecting known distances and the circle’s diameter.

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