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Worksheet
Explore the properties, theorems, and applications of circles in geometry, including tangents, chords, and angles subtended by arcs.
Circles - Practice Worksheet
Strengthen your foundation with key concepts and basic applications.
This worksheet covers essential long-answer questions to help you build confidence in Circles from Mathematics for Class X (Mathematics).
Questions
Define a tangent to a circle and explain its properties with respect to the radius at the point of contact.
Refer to Theorem 10.1 in the chapter which states the relationship between a tangent and the radius at the point of contact.
Prove that the lengths of two tangents drawn from an external point to a circle are equal.
Use the RHS congruence rule to prove the triangles formed by the tangents and the radii are congruent.
Explain the different positions a line can have with respect to a circle and define each case.
Refer to Fig. 10.1 in the chapter which illustrates the three possible positions of a line with respect to a circle.
How many tangents can be drawn from a point outside the circle, and why?
Think about the symmetry and the properties of tangents from an external point.
Describe the activity that demonstrates the existence of a tangent at a point on a circle.
Refer to Activity 1 in the chapter which involves rotating a straight wire around a point on a circular wire.
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Use the fact that the shortest distance from a point to a line is the perpendicular distance.
Find the length of a tangent drawn from a point 10 cm away from the center of a circle with radius 6 cm.
Apply the Pythagorean theorem to the right triangle formed by the radius, the tangent, and the line joining the external point to the center.
Explain why a line that is perpendicular to the radius at its outer end is a tangent to the circle.
Consider the definition of a tangent and the properties of perpendicular lines.
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the center.
Use the properties of quadrilaterals and the fact that the sum of angles in a quadrilateral is 360°.
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.
Use the property that the lengths of tangents drawn from an external point to a circle are equal.
Circles - Mastery Worksheet
Advance your understanding through integrative and tricky questions.
This worksheet challenges you with deeper, multi-concept long-answer questions from Circles to prepare for higher-weightage questions in Class X Mathematics.
Questions
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Use the property that the shortest distance from a point to a line is the perpendicular distance.
Two tangents TP and TQ are drawn to a circle with center O from an external point T. Prove that angle PTQ = 2 * angle OPQ.
Use the properties of isosceles triangles and the fact that the tangent is perpendicular to the radius at the point of contact.
PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP.
Use the Pythagorean theorem and properties of similar triangles to find the length of the tangent.
Prove that the lengths of tangents drawn from an external point to a circle are equal.
Use the RHS congruency rule to prove the triangles formed are congruent.
In two concentric circles, prove that the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
Recall that the perpendicular from the center of a circle to a chord bisects the chord.
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.
Use the property that tangents drawn from an external point to a circle are equal in length.
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the center.
Consider the sum of angles in a quadrilateral and the properties of tangents to a circle.
Prove that the parallelogram circumscribing a circle is a rhombus.
Use the property that in a circumscribed quadrilateral, the sums of the lengths of opposite sides are equal.
Two parallel tangents of a circle meet a third tangent at points A and B. Prove that angle AOB is 90°, where O is the center of the circle.
Use the congruency of triangles formed by the tangents and the properties of angles in a circle.
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC, into which BC is divided by the point of contact D, are of lengths 8 cm and 6 cm respectively. Find the lengths of sides AB and AC.
Use the property that the lengths of the two tangents drawn from an external point to a circle are equal and the formula relating the area of a triangle to its inradius and semi-perimeter.
Circles - Challenge Worksheet
Push your limits with complex, exam-level long-form questions.
The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Circles in Class X.
Questions
Prove that the lengths of tangents drawn from an external point to a circle are equal. Use this property to solve a real-life problem involving two tangents from a point to a circular park.
Think about the right angles formed by the radius and tangent, and how the hypotenuse represents the distance from the external point to the center.
A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.
Consider the chord and two radii forming an equilateral triangle for the minor arc scenario.
Two circles intersect at two points. Prove that the line joining their centers is the perpendicular bisector of the common chord.
Draw the common chord and the line joining the centers, then use congruent triangles.
In a circle of radius 5 cm, AB and AC are two chords such that AB = AC = 6 cm. Find the length of the chord BC.
Consider the perpendicular distance from the center to the chords AB and AC.
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the center.
Consider the quadrilateral formed by the two radii and the two tangents.
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC.
Let the points of contact on AB and AC be E and F, then use the fact that AE = AF, BD = BE, and CD = CF.
Prove that the parallelogram circumscribing a circle is a rhombus.
Consider the sides of the parallelogram as tangents to the circle.
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
The chord is tangent to the smaller circle, so the radius to the point of contact is perpendicular to the chord.
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.
Consider the angles subtended by the sides at the center and use the fact that the sum of angles in a quadrilateral is 360 degrees.
A point P is 13 cm from the center of a circle of radius 5 cm. Find the lengths of the tangents drawn from P to the circle and the angle between these tangents.
Use the right-angled triangle formed by the radius, tangent, and the line joining the point to the center.
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