Given that 𝐴𝐵 equals 20, find the length of line segment 𝑂𝐴.
In the figure, we have a circle which contains a right triangle. The length 𝐴𝐵, which we’re given has a length of 20 length units, is the hypotenuse of this right triangle. We’re asked to calculate the length of the line segment 𝑂𝐴, which is one of the shorter sides in the right triangle. We can use the Pythagorean theorem to help us find a missing length in a right triangle. This theorem tells us that, in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this problem, however, we want to find the length of 𝑂𝐴. But we only know the length of one of the other sides, the hypotenuse. We don’t know the length of the line segment 𝑂𝐵. But we can use the properties of the circle to help us. The line segments 𝑂𝐴 and 𝑂𝐵 are both radii of the circle because they’re both a straight line extending from the center to a point on the circumference. That means that the length of 𝑂𝐴 is equal to the length of 𝑂𝐵.
And so if we define the length of the line segment 𝑂𝐴 as 𝑥, then that means that the length of the line segment 𝑂𝐵 would also be 𝑥. And we can now apply the Pythagorean theorem. Therefore, we’ll have the hypotenuse, that’s 20 squared, equal to 𝑥 squared plus 𝑥 squared. And of course 𝑥 squared plus 𝑥 squared is simplified to two 𝑥 squared. We can also actually work out what 20 squared is. It’s 400. We can then divide both sides by two, which gives us that 200 is equal to 𝑥 squared. We can then take the square root of each side of the equation, noting that 𝑥 has to be a positive value because it’s a length. Since 200 is equivalent to 100 multiplied by two, we have the square root of 100 times two is equal to 𝑥. And when we take the square root of 100, that leaves us with 10. And so 10 root two is equal to 𝑥.
Since we defined 𝑥 to be the length of the line segment 𝑂𝐴, then we can give the answer that this length is 10 root two. And the units here would be length units.