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Find the length of a common internal tangent Play along at home!

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Two circles with radii 5 and 3 are 1(unit) apart

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Draw the Line of Centers

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Find the length of the common internal tangent.

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The radius, segment AD, is perpendicular to the internal tangent at the point of tangency.

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The radius, segment BE, is perpendicular to the internal tangent at the point of tangency.

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Now we can finish a rectangle, BEDF.

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All radii are congruent, so we can add some lengths.

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Opposite sides of a rectangle are congruent, so DF = 3

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We already know that AB = 5+1+3; AB = 9

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Angle F is a right angle; so triangle AFB is a right triangle with hypotenuse AB.

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FB 2 = AB 2 -AF 2 FB 2 = FB 2 = FB 2 = 17 FB = sqrt(17)

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The common internal tangent (segment DE) is the opposite side of the rectangle, so it is the same length as segment FB. So the length of the common internal tangent is the Square root of 17.

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