Question 4

Step 1 · Set Up the Circle and Given Line

Draw a circle with center $O$ and a given straight line $AB$.

Step 2 · Draw Perpendicular Line to AB

Draw a line passing through center $O$ perpendicular to line $AB$, meeting $AB$ at point $P$.

Step 3 · Mark Points for Tangent and Secant

Extend line $OP$ to intersect the circle at point $X$ (on the circumference) and choose any point $Y$ on $OP$ inside the circle.

Step 4 · Construct Parallel Tangent and Secant Lines

• Draw a line $CD$ through point $X$ perpendicular to $OP$. Since $CD \perp OP$ and $AB \perp OP$, $CD \parallel AB$. Because $CD$ touches the circle at only point $X$, $CD$ is a tangent.
• Draw a line $EF$ through point $Y$ perpendicular to $OP$. Since $EF \perp OP$ and $AB \perp OP$, $EF \parallel AB$. Because $EF$ passes through the interior and cuts the circle at two points, $EF$ is a secant.