If the length of 𝐸𝐶 is 10 centimetres, the length of 𝐸𝐷 is six centimetres, the length of 𝐸𝐵 is five centimetres, find the length of the line segment 𝐸𝐴.
Let’s have a look at the diagram more closely. It consists of a circle. And the points 𝐴, 𝐵, 𝐶, and 𝐷 all lie on the circle circumference. There’s then a point 𝐸, which is exterior to the circle. The lines 𝐸𝐴 and 𝐸𝐶 each intersect the circle in two places. And therefore, the name given to lines 𝐸𝐴 and 𝐸𝐶 is secants.
We’ve been given some information about the lengths of these secants or at least about the lengths of segments of them. Let’s add this information to the diagram. Firstly, the length 𝐸𝐶 is 10 centimetres. The length 𝐸𝐷 is six centimetres. And the length 𝐸𝐵 is five centimetres. It’s the length 𝐸𝐴 that we’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, that’s the part outside the circle, are equal. For the first secant, the full secant is the line 𝐸𝐴. And the external secant segment is 𝐸𝐵. So the product is 𝐸𝐴 multiplied by 𝐸𝐵. For the other secant, the full secant is the line 𝐸𝐶. And the external segment is 𝐸𝐷.
So we have an equation. 𝐸𝐴 multiplied by 𝐸𝐵 is equal to 𝐸𝐶 multiplied by 𝐸𝐷. We can substitute the values we already know. 𝐸𝐵 is five. 𝐸𝐶 is 10. And 𝐸𝐷 is six. So we have 𝐸𝐴 multiplied by five is equal to 10 multiplied by six. To solve this equation for 𝐸𝐴, we need to divide both sides by five, giving 𝐸𝐴 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 𝐸𝐴, 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 segment’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.