Question

Two circles with radii r₁ and r₂ touch each other externally. If the length of their direct common tangent is T, which of the following is the correct relationship between T, r₁, and r₂ ?

• A. T= r₁ + r₂
• B. T= 2( r₁ + r₂)
• C. T= 2√r₁r₂
• D. T= r₁² + r₂²

Answer 1

Answer:

C. T= 2√r₁r₂

If two circles of radii (r_1) and (r_2) touch externally, then the distance between their centres is

$d=r_1+r_2.$

The length (T) of the direct (external) common tangent is given by

$T=\sqrt{d^2-(r_1-r_2)^2}.$

Substituting$(d=r_1+r_2):$

$T=\sqrt{(r_1+r_2)^2-(r_1-r_2)^2}.$

Using the identity

$(a+b)^2-(a-b)^2=4ab,$

we get

$T=\sqrt{4r_1r_2}=2\sqrt{r_1r_2}.$

Answer:

$\boxed{T=2\sqrt{r_1r_2}}$

or equivalently,

$\boxed{T^2=4r_1r_2.}$