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Prove that the tangents drawn from an external point to a circle are equal in length.
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Letβs sketch a diagram.
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We can draw a circle with center π with two tangents drawn from an external point π.
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Letβs also call the points at which these tangents meet the circle π΄ and π΅.
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Adding in the line segments ππ΄ and ππ΅, we can see that we have constructed two radii for our circle.
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Since these are the radii of the same circle, we can deduce that the lines ππ΄ and ππ΅ are of equal length.
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We also know that the radius and tangents meet at 90 degrees.
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So angle ππ΄π and ππ΅π are both 90 degrees.
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Letβs add one more line into our diagram.
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Thatβs the line segment joining π to π.
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We have now created two triangles ππ΄π and ππ΅π that have a shared line ππ.
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We have already shown that the sides ππ΄ and ππ΅ are of equal length and that angle ππ΄π is equal to angle ππ΅π, which is equal to 90 degrees.
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The triangles have a right angle each, a hypotenuse of equal length, and one other side is also of equal length.
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This is a condition of congruency, often called RHS, where R stands for right angle, H stands for hypotenuse, and S stands for side.
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We can, therefore, deduce that the triangles are congruent; that is to say, their sides are all of equal length.
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Since these two triangles are congruent, it follows that π΄π must be equal in length to π΅π.
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Remember we said that the lines joining π΄ and π and π΅ and π were the tangents to the circle.
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So we have proven that the tangents drawn from a point to the circle are of equal length.