Important Properties of Transverse Common Tangents

I. The two transverse common tangents drawn to two circles are equal in length.

Given:

WX and YZ are two transverse common tangents drawn to the two given circles with centres O and P. WX and YZ intersect at T.

Equal Transverse Common Tangents

To prove: WX = YZ.

Proof:

Statement

Reason

1. WT = YT.

1. The two tangents, drawn to a circle from an external point, are equal in length.

2. XT = ZT.

2. An in statement 1.

3. WT + XT = YT + ZT

⟹ WX = YZ. (Proved)

3. Adding statements 1 and 2.

Length of a Transverse Common Tangent


II. The length of a transverse common tangent to two circles is \(\sqrt{d^{2} – (r_{1} + r_{2})^{2}}\), where d is the distance between the centres of the circles, and r\(_{1}\) and r\(_{2}\) are the radii of the given circles.

Proof:

Let two circles be given with centres O and P, and radii r\(_{1}\) and r\(_{2}\) respectively, where r\(_{1}\) < r\(_{2}\). Let the distance between the centres of the circles, OP = d.

Let WX be a transverse common tangent.

Therefore, OW = r\(_{1}\) and PX = r\(_{2}\).

Also, OW ⊥ WX and PX ⊥ WX, because a tangent is perpendicular to the radius drawn through the point of contact

Produce W to T such that WT = PX = r\(_{2}\). Join T to P. In the quadrilateral WXPT, WT ∥ PX, as both are perpendiculars to WX; and WT = PX. Therefore, WXPT is a rectangle. Thus, WX = PT, as the opposite sides of a rectangle are equal.

OT = OW + WT = r\(_{1}\)  +  r\(_{2}\).

In the right-angled triangle OPT, we have

PT2 = OP2 – OT2 (by Pythagoras’ Theorem)

⟹ PT2 = d2 – (r\(_{1}\) + r\(_{1}\))\(^{2}\)

⟹ PT = \(\sqrt{d^{2} – (r_{1} + r_{2})^{2}}\)

⟹ WX = \(\sqrt{d^{2} – (r_{1} + r_{2})^{2}}\) (Since, PT = WX).


III. The transverse common tangents drawn to two circles intersect on the line drawn through the centres of the circles.

Given: Two circles with centres O and P, and their transverse common tangents WX and YZ, which intersects at T

Properties of Transverse Common Tangents

To prove: T lies on the line joining O to P, i.e., O T and P lie on the same straight line.

Proof:

Statement

Reason

1. OT bisects ∠WTY

⟹ ∠ATO = \(\frac{1}{2}\)∠WTY.

1. The tangents drawn to a circle from an external point are equally inclined to the line joining the point to the centre of the circle.

2. TP bisects ∠ZTX

⟹ ∠XTP = \(\frac{1}{2}\)∠ZTX.

2.  As in statement 1.

3. ∠WTY = ∠ZTX.

3. Vertically opposite angles.

4. ∠WTO = ∠XTP.

4. From statement 1, 2 and 3.

5. OT and TP lie on the same straight line

⟹ O, T, P are collinear. (Prove)

5. The two angles are forming a pair of vertically opposite angles.




10th Grade Math

From Important Properties of Transverse Common Tangents 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

  1. 5th Grade Circle Worksheet | Free Worksheet with Answer |Practice Math

    Jul 11, 25 02:14 PM

    Radii of the circRadii, Chords, Diameters, Semi-circles
    In 5th Grade Circle Worksheet you will get different types of questions on parts of a circle, relation between radius and diameter, interior of a circle, exterior of a circle and construction of circl…

    Read More

  2. Construction of a Circle | Working Rules | Step-by-step Explanation |

    Jul 09, 25 01:29 AM

    Parts of a Circle
    Construction of a Circle when the length of its Radius is given. Working Rules | Step I: Open the compass such that its pointer be put on initial point (i.e. O) of ruler / scale and the pencil-end be…

    Read More

  3. Combination of Addition and Subtraction | Mixed Addition & Subtraction

    Jul 08, 25 02:32 PM

    Add and Sub
    We will discuss here about the combination of addition and subtraction. The rules which can be used to solve the sums involving addition (+) and subtraction (-) together are: I: First add

    Read More

  4. Addition & Subtraction Together |Combination of addition & subtraction

    Jul 08, 25 02:23 PM

    Addition and Subtraction Together Problem
    We will solve the different types of problems involving addition and subtraction together. To show the problem involving both addition and subtraction, we first group all the numbers with ‘+’ and…

    Read More

  5. 5th Grade Circle | Radius, Interior and Exterior of a Circle|Worksheet

    Jul 08, 25 09:55 AM

    Semi-circular Region
    A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known

    Read More