1 added
2 removed
Original
2026-01-01
Modified
2026-02-28
1
-
<p>137 Learners</p>
1
+
<p>160 Learners</p>
2
<p>Last updated on<strong>September 10, 2025</strong></p>
2
<p>Last updated on<strong>September 10, 2025</strong></p>
3
<p>Tangents to a circle have several unique properties that are essential in solving geometric problems related to circles. These properties help students analyze and solve problems involving tangents, angles, and distances. The properties include: a tangent to a circle is perpendicular to the radius at the point of tangency, and tangents drawn from a common external point are equal in length. Understanding these properties assists students in solving problems related to circles effectively. Now let us learn more about the properties of tangents to a circle.</p>
3
<p>Tangents to a circle have several unique properties that are essential in solving geometric problems related to circles. These properties help students analyze and solve problems involving tangents, angles, and distances. The properties include: a tangent to a circle is perpendicular to the radius at the point of tangency, and tangents drawn from a common external point are equal in length. Understanding these properties assists students in solving problems related to circles effectively. Now let us learn more about the properties of tangents to a circle.</p>
4
<h2>What are the Properties of Tangents to a Circle?</h2>
4
<h2>What are the Properties of Tangents to a Circle?</h2>
5
<p>The properties of tangents to a circle are fundamental and help students understand and work with this geometric concept. These properties are derived from the<a>principles of geometry</a>. There are several properties of tangents to a circle, and some of them are mentioned below:</p>
5
<p>The properties of tangents to a circle are fundamental and help students understand and work with this geometric concept. These properties are derived from the<a>principles of geometry</a>. There are several properties of tangents to a circle, and some of them are mentioned below:</p>
6
<p><strong>Property 1: Perpendicularity</strong></p>
6
<p><strong>Property 1: Perpendicularity</strong></p>
7
<p>A tangent to a circle is perpendicular to the radius at the point of tangency.</p>
7
<p>A tangent to a circle is perpendicular to the radius at the point of tangency.</p>
8
<p><strong>Property 2: Equal Tangents</strong></p>
8
<p><strong>Property 2: Equal Tangents</strong></p>
9
<p>Tangents drawn from a common external point to a circle are equal in length.</p>
9
<p>Tangents drawn from a common external point to a circle are equal in length.</p>
10
<p><strong>Property 3: Angle Between Tangents</strong></p>
10
<p><strong>Property 3: Angle Between Tangents</strong></p>
11
<p>The angle between two tangents drawn from a common external point is bisected by the line segment joining the center of the circle to the external point.</p>
11
<p>The angle between two tangents drawn from a common external point is bisected by the line segment joining the center of the circle to the external point.</p>
12
<p><strong>Property 4: Tangent-Secant</strong></p>
12
<p><strong>Property 4: Tangent-Secant</strong></p>
13
<p>The<a>square</a>of the length of a tangent drawn from an external point to a circle is equal to the<a>product</a>of the lengths of the secant segment and its external segment from the same point.</p>
13
<p>The<a>square</a>of the length of a tangent drawn from an external point to a circle is equal to the<a>product</a>of the lengths of the secant segment and its external segment from the same point.</p>
14
<h2>Tips and Tricks for Properties of Tangents to a Circle</h2>
14
<h2>Tips and Tricks for Properties of Tangents to a Circle</h2>
15
<p>Students often find the properties of tangents to a circle confusing. To avoid such confusion, we can follow these tips and tricks:</p>
15
<p>Students often find the properties of tangents to a circle confusing. To avoid such confusion, we can follow these tips and tricks:</p>
16
<p><strong>Perpendicularity to the Radius:</strong>Students should remember that a tangent to a circle is always perpendicular to the radius at the point of tangency. They can verify this by drawing a line from the circle's center to the tangent point and observing the right angle formed.</p>
16
<p><strong>Perpendicularity to the Radius:</strong>Students should remember that a tangent to a circle is always perpendicular to the radius at the point of tangency. They can verify this by drawing a line from the circle's center to the tangent point and observing the right angle formed.</p>
17
<p><strong>Equal Tangents:</strong>Students should recall that tangents drawn from a common external point are equal in length. To verify this, students can measure the tangents from the same external point to different points of tangency on the circle.</p>
17
<p><strong>Equal Tangents:</strong>Students should recall that tangents drawn from a common external point are equal in length. To verify this, students can measure the tangents from the same external point to different points of tangency on the circle.</p>
18
<p><strong>Tangent-Secant Relationship:</strong>Students should remember that the square of the tangent length equals the product of the whole secant segment and its external segment from the same external point.</p>
18
<p><strong>Tangent-Secant Relationship:</strong>Students should remember that the square of the tangent length equals the product of the whole secant segment and its external segment from the same external point.</p>
19
<h2>Confusing Tangents with Secants</h2>
19
<h2>Confusing Tangents with Secants</h2>
20
<p>Students should remember that a tangent touches the circle at one point, whereas a secant intersects the circle at two points.</p>
20
<p>Students should remember that a tangent touches the circle at one point, whereas a secant intersects the circle at two points.</p>
21
<h3>Explore Our Programs</h3>
21
<h3>Explore Our Programs</h3>
22
-
<p>No Courses Available</p>
23
<h3>Problem 1</h3>
22
<h3>Problem 1</h3>
24
<p>Using the Pythagorean theorem, since OP is the hypotenuse and OA is perpendicular, PA = √(OP² - OA²) = √(10² - 6²) = √(100 - 36) = √64 = 8 cm.</p>
23
<p>Using the Pythagorean theorem, since OP is the hypotenuse and OA is perpendicular, PA = √(OP² - OA²) = √(10² - 6²) = √(100 - 36) = √64 = 8 cm.</p>
25
<p>Okay, lets begin</p>
24
<p>Okay, lets begin</p>
26
<p>Two tangents PA and PB are drawn from an external point P to a circle with center O. If PA = 15 cm, what is the length of PB?</p>
25
<p>Two tangents PA and PB are drawn from an external point P to a circle with center O. If PA = 15 cm, what is the length of PB?</p>
27
<p>Well explained 👍</p>
26
<p>Well explained 👍</p>
28
<h3>Problem 2</h3>
27
<h3>Problem 2</h3>
29
<p>Since tangents drawn from a common external point are equal, PA = PB. Thus, PB = 15 cm.</p>
28
<p>Since tangents drawn from a common external point are equal, PA = PB. Thus, PB = 15 cm.</p>
30
<p>Okay, lets begin</p>
29
<p>Okay, lets begin</p>
31
<p>In a circle, a tangent at point A meets a line segment OP where O is the center of the circle. If OA = 5 cm and OP = 13 cm, find the length of PA.</p>
30
<p>In a circle, a tangent at point A meets a line segment OP where O is the center of the circle. If OA = 5 cm and OP = 13 cm, find the length of PA.</p>
32
<p>Well explained 👍</p>
31
<p>Well explained 👍</p>
33
<h3>Problem 3</h3>
32
<h3>Problem 3</h3>
34
<p>Using the Pythagorean theorem, since OP is the hypotenuse and OA is perpendicular, PA = √(OP² - OA²) = √(13² - 5²) = √(169 - 25) = √144 = 12 cm.</p>
33
<p>Using the Pythagorean theorem, since OP is the hypotenuse and OA is perpendicular, PA = √(OP² - OA²) = √(13² - 5²) = √(169 - 25) = √144 = 12 cm.</p>
35
<p>Okay, lets begin</p>
34
<p>Okay, lets begin</p>
36
<p>If a tangent from a point P to a circle is 9 cm long and the distance from P to the center of the circle is 15 cm, what is the radius of the circle?</p>
35
<p>If a tangent from a point P to a circle is 9 cm long and the distance from P to the center of the circle is 15 cm, what is the radius of the circle?</p>
37
<h3>Explanation</h3>
36
<h3>Explanation</h3>
38
<p>Radius = 12 cm.</p>
37
<p>Radius = 12 cm.</p>
39
<p>Well explained 👍</p>
38
<p>Well explained 👍</p>
40
<h3>Problem 4</h3>
39
<h3>Problem 4</h3>
41
<p>Using the Pythagorean theorem, since OP is the hypotenuse and PA is perpendicular, Radius = √(OP² - PA²) = √(15² - 9²) = √(225 - 81) = √144 = 12 cm.</p>
40
<p>Using the Pythagorean theorem, since OP is the hypotenuse and PA is perpendicular, Radius = √(OP² - PA²) = √(15² - 9²) = √(225 - 81) = √144 = 12 cm.</p>
42
<p>Okay, lets begin</p>
41
<p>Okay, lets begin</p>
43
<p>A circle has center O, and two tangents PA and PB are drawn from point P such that PA = 6 cm. If angle APB = 60 degrees, what is angle AOB?</p>
42
<p>A circle has center O, and two tangents PA and PB are drawn from point P such that PA = 6 cm. If angle APB = 60 degrees, what is angle AOB?</p>
44
<h3>Explanation</h3>
43
<h3>Explanation</h3>
45
<p>Angle AOB = 120 degrees.</p>
44
<p>Angle AOB = 120 degrees.</p>
46
<p>Well explained 👍</p>
45
<p>Well explained 👍</p>
47
<h2>A tangent to a circle is a straight line that touches the circle at exactly one point.</h2>
46
<h2>A tangent to a circle is a straight line that touches the circle at exactly one point.</h2>
48
<h3>1.How many tangents can be drawn from a point outside a circle?</h3>
47
<h3>1.How many tangents can be drawn from a point outside a circle?</h3>
49
<p>Two tangents can be drawn from a point outside a circle.</p>
48
<p>Two tangents can be drawn from a point outside a circle.</p>
50
<h3>2.Are tangents from the same external point equal?</h3>
49
<h3>2.Are tangents from the same external point equal?</h3>
51
<p>Yes, tangents drawn from the same external point to a circle are equal in length.</p>
50
<p>Yes, tangents drawn from the same external point to a circle are equal in length.</p>
52
<h3>3.How do you find the length of a tangent?</h3>
51
<h3>3.How do you find the length of a tangent?</h3>
53
<p>To find the length of a tangent, you can use the Pythagorean theorem if the radius and the distance from the external point to the center of the circle are known.</p>
52
<p>To find the length of a tangent, you can use the Pythagorean theorem if the radius and the distance from the external point to the center of the circle are known.</p>
54
<h3>4.Can a tangent pass through the circle?</h3>
53
<h3>4.Can a tangent pass through the circle?</h3>
55
<p>No, a tangent can only touch the circle at one point and cannot pass through it.</p>
54
<p>No, a tangent can only touch the circle at one point and cannot pass through it.</p>
56
<h2>Common Mistakes and How to Avoid Them in Properties of Tangents to a Circle</h2>
55
<h2>Common Mistakes and How to Avoid Them in Properties of Tangents to a Circle</h2>
57
<p>Students tend to get confused when understanding the properties of tangents to a circle, and they often make mistakes while solving problems related to these properties. Here are some common mistakes and solutions to avoid them.</p>
56
<p>Students tend to get confused when understanding the properties of tangents to a circle, and they often make mistakes while solving problems related to these properties. Here are some common mistakes and solutions to avoid them.</p>
58
<p>What Is Geometry? 📐 | Shapes, Angles & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math</p>
57
<p>What Is Geometry? 📐 | Shapes, Angles & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math</p>
59
<p>▶</p>
58
<p>▶</p>
60
<h2>Hiralee Lalitkumar Makwana</h2>
59
<h2>Hiralee Lalitkumar Makwana</h2>
61
<h3>About the Author</h3>
60
<h3>About the Author</h3>
62
<p>Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.</p>
61
<p>Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.</p>
63
<h3>Fun Fact</h3>
62
<h3>Fun Fact</h3>
64
<p>: She loves to read number jokes and games.</p>
63
<p>: She loves to read number jokes and games.</p>