Okay, so now this is theorem one. We are given that OP is perpendicular to AB, right? And O is the center. So we are required to prove that AP is equal to PB.
So from here, AP is equal to PB. So we are required to prove that. So the first thing we're going to do is we're going to construct, construct line OA. So meaning that we're going to construct line OA there, and we also going to construct line OB. So going to construct line OB.
So now, now that you constructed these lines, we have formed triangle AOP and triangle OBP. So this is the first triangle, this is the second triangle, right? So now, because we are required to prove that AP is equal to PB, so you're going to use congruency to prove that. Going to use congruency using these triangles.
So the first thing you're going to do is you're going to say P1 which is this one here, is equals to P2. So angle P1 is equals to angle P2, right? So reason for that would be given because you were told that OP is perpendicular to AB. And then another thing that we can identify in these two triangles is that they share a common side.
OP is equal to OP, right? So the reason for that is common, common side. And then the third thing that we can prove in this triangle here is that this line of OA is equal to OB.
So if we check OA and OB, they are the radius from the center to the circumference. and from the circumference to the center. So they are equal. So this side here is equal to that side here.
So you're just going to write ready. So now that you've proven three, three things, you've proven the angle, you've proven the side and you've proven the side. So you're going to conclude and say, try and now a O P is congruent to try and O B P. So because.
there's a right angle, try right angle that you used to prove this thing here. So you're just going to say right hypotenuse side. So the right hypotenuse side, this R means the right angle, this right angle here, which is 90 degrees, right? And then the H means hypotenuse, meaning that we've proven the side of the hypotenuse, which is the radii. right?
And then the side, which is the common side. So this is the common, common side. And this is the radii. And then this is the 90 degrees, right?
So that's why we said right hypotenuse side. So now, therefore, we can conclude by saying AP is equals to PB. Why? Because The triangle of AOP is congruent to the triangle of OBP with this reason here.