Construction of Tangents and Chord for a Circle

overview

In this page, a short overview of the following are provided.

• Construction of a Tangent on a Point on a Circle

• Construction of Tangents from a Point outside a Circle

• Construction of chords of given length on a Circle

• Construction of chords at a given distance from center of a Circle

tangent to a circle

Consider the circle centered at $O$. The objective is to construct a tangent on the given point $P$ on the circle.

To achieve that, we use the known property "tangent is at right angle $90}^{\circ$ to the radius at the point".

Considering the construction of tangent on the given point $P$ on a circle.

It is noted that the tangent is at right angle to the radius at the point. The ray $\overrightarrow{OP}$ is constructed. Construct a perpendicular on point $P$ on the ray $\overrightarrow{OP}$. This construction problem is modified to the known problem : construction of a perpendicular on a point in a line.

The following steps detail construction of tangent on point $P$ on a circle:

• The ray $\overrightarrow{OP}$ is constructed

• On the point $P$, the perpendicular to $\overrightarrow{OP}$ is constructed. *(Construction of a perpendicular on a point is explained in the earlier lessons)*

The tangent is constructed.

summary

**Construction of Tangent on a point on circle**: Using the property that the radius and tangent on a point are at right angle, the construction is modified to construction of perpendicular to a line on a point.

tangent from outside

Consider the circle centered at $O$. The objective is to construct a tangent from the given point $P$ outside the circle. To achieve that we use the property, tangent is at right angle $90}^{\circ$ to the radius at the point of tangent.

Considering the construction of tangent from the given point $P$ outside the circle.

The point of tangent on the circle is visualized as point $A$. It is shown in the figure to illustrate. The point $A$ is not yet marked. It is noted that $\u25b3OAP$ is a right angled triangle with $\overline{OP}$ as the hypotenuse.

To mark the point $A$, "constructing a circle with hypotenuse as diameter".

*The point $A$ is not yet marked.* It is noted that $\u25b3OAP$ is a right angled triangle with $\overline{OP}$ as the hypotenuse.

The angle subtended by a diameter on the circle is right-angled.

Using this knowledge, It is visualized that a circle with diameter $\overline{OP}$ intersects the given circle at point $A$, where angle $\angle OAP$ is $90}^{\circ$ and so, $\overrightarrow{PA}$ is the tangent at point $A$.

The following steps detail construction of tangent from point $P$ outside a circle:

• The ray $\overrightarrow{OP}$ is constructed

• A circle is constructed with $\overline{OP}$ as the diameter. *(Construction of a circle with a given line segment as diameter is explained in the earlier lessons)*

• The points of intersection of the given circle to the constructed circle are connected to $P$ as the tangents of the given circle.

The tangents are constructed.

summary

**Construction of Tangents from a point outside a Circle** : Note the following properties

• the radius and tangent are perpendicular,

• the angle subtended by diameter on a circle is right-angle

the line segment between the center of the circle and the given point is considered as diameter.

The construction is modified to construction of a circle on a given line segment as diameter.

chord of length

Consider the circle centered at $O$. The objective is to construct a chord of length $\overline{PQ}=4$cm on the given point $P$ on the circle.

To achieve that a compass is enough. A compass constructs equidistant points.

The following steps detail construction of chord at point $P$ on a circle:

• Use a compass to construct an arc of length $4$ cm.

• The point of intersection of the arc on the circle is marked $Q$.

• Connect the points $P$ and $Q$

The chord $\overline{PQ}$ is constructed.
*Note: There are two chords possible from the given point $P$ of the given length.*

summary

**Construction of Chords of given length** : An arc of the given length is used to locate the points on circle and the chord is constructed.

chord at distance

Consider the circle centered at $O$. The objective is to construct a chord at $2$cm distance from the center. To achieve that we use the definition : the distance of chord from the center is length of the perpendicular line from the center.

The following steps detail construction of chord at a distance:

• Construct a ray $\overrightarrow{OR}$.

• Using a compass, measure the given $2$cm distance and mark on the ray $\overrightarrow{OR}$.

• At the point of intersection, construct a perpendicular $\overrightarrow{PQ}$

• The ray intersects at points $P$ and $Q$

The chord $\overline{PQ}$ is constructed.

summary

**Construction of Chord at a distance from Center of the circle** : Using the property that the distance of a chord from the center is measured as the perpendicular distance, a perpendicular is constructed on a radius at the given distance.

Outline

The outline of material to learn "Consrtruction (High school)" is as follows.

Note: *Click here for detailed overview of "Consrtruction (High school)" *

Note 2: * click here for basics of construction, which is essential to understand this. *

→ __Construction of Triangles With Secondary Information__

→ __Construction: Scaling a line, triangle, polygon, circle__

→ __Construction of Tangents and Chord for a circle__