Video Transcript
If the length of ๐ธ๐ถ is 10 centimetres, the length of ๐ธ๐ท is six centimetres, the length of ๐ธ๐ต is five centimetres, find the length of the line segment ๐ธ๐ด.
Letโs have a look at the diagram more closely. It consists of a circle. And the points ๐ด, ๐ต, ๐ถ, and ๐ท all lie on the circle circumference. Thereโs then a point ๐ธ, which is exterior to the circle. The lines ๐ธ๐ด and ๐ธ๐ถ each intersect the circle in two places. And therefore, the name given to lines ๐ธ๐ด and ๐ธ๐ถ is secants.
Weโve been given some information about the lengths of these secants or at least about the lengths of segments of them. Letโs add this information to the diagram. Firstly, the length ๐ธ๐ถ is 10 centimetres. The length ๐ธ๐ท is six centimetres. And the length ๐ธ๐ต is five centimetres. Itโs the length ๐ธ๐ด that weโve 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, thatโs the part outside the circle, are equal. For the first secant, the full secant is the line ๐ธ๐ด. And the external secant segment is ๐ธ๐ต. So the product is ๐ธ๐ด multiplied by ๐ธ๐ต. For the other secant, the full secant is the line ๐ธ๐ถ. And the external segment is ๐ธ๐ท.
So we have an equation. ๐ธ๐ด multiplied by ๐ธ๐ต is equal to ๐ธ๐ถ multiplied by ๐ธ๐ท. We can substitute the values we already know. ๐ธ๐ต is five. ๐ธ๐ถ is 10. And ๐ธ๐ท is six. So we have ๐ธ๐ด multiplied by five is equal to 10 multiplied by six. To solve this equation for ๐ธ๐ด, we need to divide both sides by five, giving ๐ธ๐ด 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 ๐ธ๐ด, 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 segmentโs 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.
