When two segments from same exterior point are tangent to a circle then the two segments are congruent?

2. If two segments from the from the same exterior point are tangent to a circle, then the two segments are congruent. Given: and are tangent to and and , respectively Prove:

Step-by-step answer:

We know that both triangles SME and SLE share a hypotenuse, which is line SE.

Since, we know that the definition of tangent to a circle mean that a line segment (in this case EM and EL) form a 90 degree angle with the radius, meaning that the base of the triangles are the radius.

From the Pythagorean Theorem, sqrt(a^2 + b^2) = c, where c is the hypotenuse. Since, they share a hypotenuse, they have the same c. Now, let a be the base, which is r, the other side be b, and the hypotenuse be h, so substitute:

Triangle SME:

r^2 + b^2 = h^2

b = sqrt(r^2 + h^2)

Triangle SLE:

r^2 + b^2 = h^2

b = sqrt(r^2 + h^2)

So in conclusion:

Triangle SME:

base = r                    (given)

hypotenuse = h       (given)

other side = sqrt(r^2 + h^2)

Triangle SLE:

base = r                    (given)

hypotenuse = h       (given)

other side = sqrt(r^2 + h^2)

Therefore, by definition, triangles with same sides are congruent, so triangles SME and SLE are congruent since their sides are the same. The base, hypotenuse, and other sides are r, h, and sqrt(r^2 + h^2), respectively.