一条线可以通过三种方式穿过一个圆-它可以仅穿过圆的单个点，与圆相交，也可以穿过圆的外部。穿过圆的单个点的线称为切线。当一条线与圆在两个点处相交时，我们将该线称为切线。建筑师和设计师每天在工作中都使用这个概念。在这种情况下，必须知道如何从圆外的点构造与圆的切线。让我们详细看一下该过程。
切线
圆的切线是在单个点处与圆相交的线。下图显示了切线通过圆的示例。切线与圆相交的点称为接触点。请注意，切线也垂直于连接与圆接触点的线。
Note: The lines which intersect circles at two points are called secants.
切线的构造
可以从圆外的任何点绘制两个切线。我们的目标是学习如何从圆外给出的任何点构造单个切线。可以以类似方式绘制其他切线。
施工步骤：
我们给了一个圆心，圆心为O，其半径为“ r”，圆外为A点。让我们看看如何构造从点A到圆的切线。
步骤1.绘制一个半径为r的圆，将其称为中心O。在圆外绘制一个点A。
步骤2.加入OA并将其一分为二。我们将其称为中点M。
步骤3.现在以M为中心，以MA为半径。在点B和C处绘制与给定圆相交的圆。
步骤4.加入AB和AC。它们是我们给定的切线。
样本问题
问题1：从距半径5cm的圆的中心O约10cm的点P处切线。
解决方案：
Circle with centre O with a radius “5cm” and a point P outside the circle.
Step 1. Draw a circle with radius 5cm and let’s call its centre O. Draw a point A which is 10cm from the centre outside the circle.
Step 2. Join OP and bisect it. We’ll call its midpoint M.
Step 3. Now with M as centre and radius MP. Draw a circle that intersects the given circle at points Q and R.
Step 4. Join PQ and PR. They are our given tangents.
问题2：从距半径3cm的圆的中心O 6cm的点P处切线。
解决方案：
Circle with centre O with a radius “3cm” and a point P outside the circle.
Step 1. Draw a circle with radius 3cm and let’s call its centre O. Draw a point A which is 6cm from the centre outside the circle.
Step 2. Join OP and bisect it. We’ll call its midpoint M.
Step 3. Now with M as centre and radius MP. Draw a circle that intersects the given circle at points Q and R.
Step 4. Join PQ and PR. They are our given tangents.
问题3：一个圆中可以有多少个平行切线?
回答：
There are at max two possible parallel tangents in a circle. They lie on diametrically opposite points.
问题4：从距半径5cm的圆的中心O 15cm处的点P处切线。
解决方案：
Circle with centre O with a radius “15cm” and a point P outside the circle.
Step 1. Draw a circle with radius 5cm and let’s call its centre O. Draw a point A which is 15cm from the centre outside the circle.
Step 2. Join OP and bisect it. We’ll call its midpoint M.
Step 3. Now with M as centre and radius MP. Draw a circle that intersects the given circle at points Q and R.
Step 4. Join PQ and PR. They are our given tangents.
问题5：从与该点和半径为8cm的圆心相连的线倾斜30°的点P处切线。
解决方案：
Circle with centre O with a radius “8cm” and a point P outside the circle.
We know that the tangents are inclined at 30°. Let’s say they intersect the circle at R and Q. It makes a triangle OPR. We need to know the length of OP. We also know that OR is perpendicular to PR. Thus, it’s a right-angled triangle, and we can use trigonometric formulas to find out OP.
Thus, OP is 16cm. Now we can construct a tangent.
Step 1. Draw a circle with radius 8cm and let’s call its centre O. Draw a point P which is 16cm from the centre outside the circle.
Step 2. Join OP and bisect it. We’ll call its midpoint M.
Step 3. Now with M as centre and radius MP. Draw a circle that intersects the given circle at points Q and R.
Step 4. Join PQ and PR. They are our given tangents.
问题6：从半径6cm的圆的中心O到11cm的点P处绘制切线。
解决方案：
Circle with centre O with a radius “6cm” and a point P outside the circle.
Step 1. Draw a circle with radius 6cm and let’s call its centre O. Draw a point A which is 11cm from the centre outside the circle.
Step 2. Join OP and bisect it. We’ll call its midpoint M.
Step 3. Now with M as centre and radius MP. Draw a circle that intersects the given circle at points Q and R.
Step 4. Join PQ and PR. They are our given tangents.
