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We are given the length of one side of this triangle, ? ? = 9 6 c m. Using the Pythagorean theorem, we can write The length of the line ? ?, which we are looking for, is the hypotenuse of a right triangle. This tells us that ∠ ? ? ? is a right angle With the tangent at the point of contact. We are given that ⃖ ⃗ ? ? is tangent to the circle centered at ? at ?,Īnd we can see that ? ? is a radius of the circle centered at ?, intersecting
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We can determine ∠ ? ? ? by recalling that a tangent to a circle is perpendicular The length, ? ?, we are looking for is a side length in triangle ? ? ?, so we begin by determiningĪn angle in this triangle. In our first example, we will use this theorem to find an unknown length in a diagram involving a circle and a tangent. Of the circle to the tangent, it must be perpendicular to the tangent. Since the radius is the shortest line segment connecting the center Other points on the tangent lie outside of the circle. Hence, the radius must be the shortest distance between the center of the circle and the tangent, since all On the other hand, the distance between the center of a circle and the point of contact is the radius We know that the distance between the center of a circle and a point exterior to the circle must be greater than the If a line is tangent to a circle, any point of the line is outside the circle, except the point of contact that lies on theĬircle. In other words, the shortest line segment from the point to the given line must intersect The proof of this theorem relies on the fact that the shortest distance between a line and a point is the perpendicularĭistance between the two objects.