Construction of Tangents to a Circle Practice Problems

Tangents to a circle and their applications are important and versatile concepts in geometry important in real-world applications such as in engineering, construction, and architecture. This knowledge is significant for students since constructing tangents is a problem that enhances the spatial and logical presentation of many subjects.

The purpose of this article is to introduce the reader to the background information on the topic at hand, discuss several problems and provide solutions for better practice.

What is Construction of Tangents to a Circle?

A line that comes in contact with a circle at one single point is an example of a tangent to the circle at that point called the point of tangency. This line is perpendicular to the drawn radius or extends up to the point of tangency. Constructing tangents to a circle is the concept that widens up the ways of presenting a particular line from a definite point either inside or outside the circle.

Important Formulas / Concepts

• Radius-Tangent Theorem: Tangent to a circle is a line that is drawn through such that it is perpendicular to the radius drawn at the point of contact.

• Two Tangent Theorem: External bisectors of the angles formed by a tangent and a radius are parallel to each other and have equal lengths.

• Tangents from a Point on the Circle: Circles also come in with the following important statement; only one tangent can be given at a point on the circle.

• Tangents from a Point Outside the Circle: We can draw two starting from a particular point at least for a circle that has been drawn.

Construction of Tangents to a Circle: Practice Questions

Q1. Draw a tangent from a point P which is 10cm from the centre O of the circle of radius 5cm. 

Solution:

Circle with centre O with a radius “5 cm” and a point P outside the circle.

Step 1: Draw a circle with a radius of 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 the required tangents. 

Q2. Construct a tangent to a circle from a point on the circle.

Solution:

Step 1: Place two points O and O’ on the plane where O is the origin and r is the radius of the circle to be drawn.

Step 2: Choose a point P that lies on the circle.

Step 3: Draw the radius OP.

Step 4: At point P, using a protractor, start by drawing a line perpendicular to OP. This line is tangent to the circle at point P.

Q3. Construct tangents to a circle from a point outside the circle.

Solution:

Step 1: Draw a circle with centre O and radius r where you want to place your n-ary tree or structure.

Step 2: With a point, mark a point P outside the circle.

Step 3: O and P are two points: join O and P with a straight line.

Step 4: Determine the point M midline of OP and sketch a circle with M as its centre and with point M as the terminal point of its radius. Let this circle cut the first circle at points A and B.

Step 5: Join point A and point B by a straight line and label it PA and PB. These are the lines touching the circle with passing through the point P.

Q4. Construct a tangent to a circle from a point on the circle using an alternate method.

Solution:

Step 1: Draw a circle with center O and radius r.

Step 2: Mark a point P on the circle.

Step 3: Draw the radius OP.

Step 4: Construct a perpendicular line to OP at P using a compass. Place the compass at P, and draw an arc that intersects the circle at two points. Connect these points with a straight line. This line is the tangent at P.

Q5. Given a circle with center O and a point P outside it, construct the tangents to the circle and prove their lengths are equal.

Solution:

Step 1: Draw a circle with center O and radius r.

Step 2: Mark a point P outside the circle.

Step 3: Connect O and P, and find the midpoint M of OP.

Step 4: Draw a circle with center M and radius MO. Let this circle intersect the original circle at points A and B.

Step 5: Draw lines PA and PB. These lines are the tangents to the circle from point P.

Step 6: To prove PA = PB, use the Pythagorean theorem in the triangles formed by the radius and the tangent segments.

Q6. Construct a pair of tangents from a point to two given circles of different radii.

Solution:

Step 1: Draw two circles with centers O₁ and O₂ and radii r₁ and r₂, respectively.

Step 2: Mark a point P outside both circles.

Step 3: Connect P to both O₁ and O₂, finding the midpoints M₁ and M₂ of PO₁ and PO₂.

Step 4: Draw circles with centers M₁ and M₂ and radii M₁O₁ and M₂O₂. Let these circles intersect the original circles at points A₁, B₁ and A₂, B₂, respectively.

Step 5: Draw lines PA₁, PB₁, PA₂, and PB₂. These lines are the tangents from P to both circles.

Q7. Construct the common tangents to two given circles.

Solution:

Step 1: Draw two circles with centers O₁ and O₂ and radii r₁ and r₂, respectively.

Step 2: Identify the distance between the centers O₁ and O₂.

Step 3: Depending on the distance, construct the external or internal tangents:

• External Tangents: If the circles do not intersect, draw lines connecting points of tangency outside both circles.
• Internal Tangents: If the circles intersect, draw lines connecting points of tangency inside both circles.

Q8. Construct the tangent to a circle using the direct method.

Solution:

Step 1: Draw a circle with center O and radius r.

Step 2: Choose a point T on the circle.

Step 3: Place the compass at O, with the radius slightly larger than OT, and draw an arc intersecting the circle at two points.

Step 4: Draw the perpendicular bisector of the segment connecting these intersection points.

Step 5: This bisector is the tangent at T.

Q9. Construct the tangent to a circle through a given external point using the circle inversion method.

Solution:

Step 1: Draw a circle with center O and radius r.

Step 2: Choose an external point P.

Step 3: Draw the power circle of P with respect to the given circle.

Step 4: Identify the points of intersection of this circle with the given circle.

Step 5: Draw the tangents from P through these intersection points.

Q10. Construct the tangents to a circle using coordinate geometry.

Solution:

Step 1: Given a circle with center (h, k) and radius r.

Step 2: Identify the external point P(a, b).

Step 3: Use the formula y = mx + c where m = − (x₁ - h )/(y₁ - k).

Step 4: Calculate the slope and the y-intercept of the tangent line.

Step 5: Draw the tangent line using these values.

Construction of Tangents to a Circle: Worksheet

P1. Draw a line parallel to the given line through a point distant 5cm from the center O of the circle.

P2. Draw the tangents passing through the point, which is 8 cm away from the circle having a radius of 4 cm.

P3. Let centre of the circle be O Then draw a tangent from a point which is at a distance of 6 cm from the centre of the circle.

P4. Draw two circles externally such that one circles diameter is 3 cm and the other circles is of 5 cm and the distance between the centers of the circles is 10 cm.

P5. Make internal tangents of two circles having radii as 4cm and 6cm having centres 12cm apart.

P6. Explain how to construct the tangent to a circle using the direct method, using a point on the circle.

P7. Drawing tangents from two intersection points that are given when two circles are intersecting.

P8. Draw coordinate geometry of tangent to a circle from a point (3, 4).

P9. Using the method of circle inversion, build the tangent from the external point having the distance of 8 cm from the center of the mentioned circle with the radius of 6 cm.

P10. Draw the tangent to a circle taking the equation (x − 2)² + (y − 3)² = 25 using analytical geometry from point (5, 7).