Angles between the Tangent and the Chord

Here we will prove that if a line touches a circle and from the point of contact a chord is down, the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternate segments.

Given: A circle with centre O. Tangent XY touches the circle at the point M. Through M, a chord MN is drawn. Let MN subtend ∠MSN and ∠MTN in the major and minor segments respectively.

Angles between the Tangent and the Chord

To prove: ∠NMY = ∠MSN and ∠NMX = ∠MTN

Construction: Draw the diameter MOR. Join N to R.

Proof:

Statement:

Reason

1. ∠RMY = 90°

⟹ ∠RMN + ∠NMY = 90°

⟹ ∠NMY = 90° - ∠RMN

1. Diameter ⊥ Tangent.

2. In ∆RMN, ∠MNR = 90°

2. Angle in a semicircle is 90°.

3. ∠NRM + ∠RMN = 90°

3. In a right-angled triangle, sum of the two acute angles is 90°.

4. ∠NRM = ∠MSN

4. Angles in the same segment are equal.

5. ∠MSN + ∠RMN = 90°

⟹ ∠MSN = 90° - ∠RMN

5. From statements 3 and 4.

6. ∠NMY = ∠MSN

6. From statements 1 and 5.

7. ∠NMY + ∠NMX = 180°

7. Linear pair.

8.  ∠MSN + ∠MTN = 180°

8. Opposite angles of a cyclic quadrilateral are supplementary.

9. ∠NMY + ∠NMX = ∠MSN + ∠MTN

9. From 7 and 8.

10. ∠NMX = ∠MTN.

10. ∠NMY = ∠MSN from statement 6.





10th Grade Math

From Angles between the Tangent and the Chord to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

RSS

  1. Counting Numbers from 1 to 50 | Match the Number | Missing Numbers

    Apr 04, 25 03:46 PM

    Math Coloring Pages on Counting Number Oredr
    In counting numbers from 1 to 50, recognize the numbers, count and then join the numbers in the correct number order. Here we mainly need eye-hand coordination to draw the picture and maintain the num

    Read More

  2. Counting Eleven to Twenty with Numbers and Words |Numbers from 11 - 20

    Apr 04, 25 03:21 PM

    Counting eleven to twenty with numbers and words are explained below. One ten and one more is eleven. Eleven comes after ten. One ten and two more is twelve. Twelve comes after eleven.

    Read More

  3. 5th Grade BODMAS Rule Worksheet | PEMDAS | Order of operations|Answers

    Apr 03, 25 03:11 PM

    5th Grade BODMAS Rule Worksheet
    In 5th Grade BODMAS Rule Worksheet you will get different types of problems on mathematical expressions involving different operations, mathematical expression with 'brackets' and 'of' and simplifying…

    Read More

  4. Worksheet on Simplification | Simplify Expressions | BODMAS Questions

    Apr 03, 25 02:58 PM

    Worksheet on Simplification
    In worksheet on simplification, the questions are based in order to simplify expressions involving more than one bracket by using the steps of removal of brackets. This exercise sheet

    Read More

  5. Divisible by 2 Video |Test of Divisibility by 2 Trick| Rules| Examples

    Apr 03, 25 10:25 AM

    Divisible by 2
    A number is divisible by 2 if the digit at unit place is either 0 or multiple of 2. So a number is divisible by 2 if digit at its units place is 0, 2, 4, 6 or 8.

    Read More