Video: Finding an Unknown Lengths of a Proportion Resulting from Two Circle Secants Drawn from the Same External Point

If ๐ธ๐ถ = 10 cm, ๐ธ๐ท = 6 cm, ๐ธ๐ต = 5 cm, find the length of the line segment ๐ธ๐ด.

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Video Transcript

If the length of ๐ธ๐ถ is 10 centimetres, the length of ๐ธ๐ท is six centimetres, the length of ๐ธ๐ต is five centimetres, find the length of the line segment ๐ธ๐ด.

Letโ€™s have a look at the diagram more closely. It consists of a circle. And the points ๐ด, ๐ต, ๐ถ, and ๐ท all lie on the circle circumference. Thereโ€™s then a point ๐ธ, which is exterior to the circle. The lines ๐ธ๐ด and ๐ธ๐ถ each intersect the circle in two places. And therefore, the name given to lines ๐ธ๐ด and ๐ธ๐ถ is secants.

Weโ€™ve been given some information about the lengths of these secants or at least about the lengths of segments of them. Letโ€™s add this information to the diagram. Firstly, the length ๐ธ๐ถ is 10 centimetres. The length ๐ธ๐ท is six centimetres. And the length ๐ธ๐ต is five centimetres. Itโ€™s the length ๐ธ๐ด that weโ€™ve been asked to find. So we need to recall the relationship that exists between the lengths of segments of secants.

We know that if two secants intersect outside a circle, then the products of the measures of each secant segment and its external secant segment, thatโ€™s the part outside the circle, are equal. For the first secant, the full secant is the line ๐ธ๐ด. And the external secant segment is ๐ธ๐ต. So the product is ๐ธ๐ด multiplied by ๐ธ๐ต. For the other secant, the full secant is the line ๐ธ๐ถ. And the external segment is ๐ธ๐ท.

So we have an equation. ๐ธ๐ด multiplied by ๐ธ๐ต is equal to ๐ธ๐ถ multiplied by ๐ธ๐ท. We can substitute the values we already know. ๐ธ๐ต is five. ๐ธ๐ถ is 10. And ๐ธ๐ท is six. So we have ๐ธ๐ด multiplied by five is equal to 10 multiplied by six. To solve this equation for ๐ธ๐ด, we need to divide both sides by five, giving ๐ธ๐ด is equal to 10 multiplied by six over five. 10 multiplied by six is 60. And 60 divided by five is 12. So the length of the line segment ๐ธ๐ด, which will have unit centimetres as these were the units given for all the other lengths in the question, is 12 centimetres.

We answered this question by recalling the secant segmentโ€™s theorem, which tells us that if two secants intersect outside a circle, then the products of the measures of each secant segment and its external secant segment are equal.

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