We’re on problem 71. It says, what is the value of x in the triangle below? Okay. So we could just pull out the Pythagorean Theorem here. And although you might recognize that if these two sides are the same and then these two base angles are going to be the same. And if those base angles are the same, then this is 90, so then, you have 90 degrees to go between those two angles. So they’re going to have to be 45, because they’re the same, right? So, there’s a 45-45-90 triangle. And if you have it already, you’ll eventually memorize kind of how the sides of a 45-45-90 relate to its hypotenuse. But you don’t have to memorize if you can prove it here. Sometimes, it’s just faster on standardized tests and things like that.
So, what does the Pythagorean Theorem tells us? It tells us that this side squared – that so let’s say x squared, plus this side squared, plus x squared is equal to the hypotenuse squared, is equal to 10 squared, which is 100. So we get 2x squared is equal to 100x squared, is equal to 5, right, dividing both sides by 2. And then, what does this turn into? So we can say x is equal to the square root of 50. Let’s see. Is there anything that we can do here to simplify it all? Let me think. X is equal to the square root of 50. Oh, sure. 50 is – so x is equal to the square root of – 50 is 25 times 2, right? So that’s equal to the square root of 25 times the square root of 2, which is equal to 5 times the square root of 2. Choice B.
Problem 72. What is the value of x in inches? Okay. So once again, a couple of problems ago, we solved 30-60-90 triangle. This is another one. 30 degrees, 90 degrees, they have to add up to 180. This one is equal to 60 degrees. And we kind of – I did that big convoluted drawing where I flipped it and all of that. And I think this is a good time to memorize the sides of a 30-60-90 triangle, because that’s something that one needs to know in life. It’s surprisingly useful especially once you start taking standardized test or do Trigonometry. So I’ll just give you the general rule. So, let me just – I’m drawing another one right here. So let’s say this is my other 30-60-90 triangle, right. This is clearly the hypotenuse up here. That’s the hypotenuse. This is the – I call it the 30 degrees side. It’s opposite the 30-degree angle or it’s the shortest side. So, the general rule is, if this side right here is x, then the hypotenuse is going to be 2x. And we saw that in the previous video. And then you can actually use a Pythagorean Theorem here to solve for this last side. You really just have to memorize that the hypotenuse is twice the shortest side. So in this case, what’s the shortest side? It’s opposite the 30 degrees side, so it’s 7. So the hypotenuse would be twice that which is 14. And you could use the Pythagorean Theorem to figure out x now. Or you could just memorize that the middle side, I guess you could say, or the long non-hypotenuse side, or the 60 degrees side, the side opposite the 60-degree angle, that’s equal to the square root of 3 times the short side. So in this case, x is a square root of 3 times 7. So, x is equal to 7 times the square root of 3. And don’t take my word for it, or you could take my word for it that this is double that and we proved that in a couple of videos ago. But you could do that Pythagorean Theorem here. You could say that 7 squared, which is 49, plus x squared, is going to be equal to the hypotenuse squared. 14 squared is what? 140, 4 times 14 is, 196. Subtract 49 from both sides. You get x squared is equal to –let’s see, 196 minus 40 would be 156. So that’s 157, is that right? Let me show you I got it. 14 times 14 – 4 times 4 is 16. 0 – 140, right, 196. And then if you would subtract 49 from that, minus 49 – let’s say this is 8. This is 16. We have a 7. Sorry, 147. it’s good thing I checked that. 147, all right. So x is equal to the square root of 147. 147 is 49 times 3, is equal to the square root of 49 times 3. Well, that’s just equal to the square root of 49 times the square root of 3, which is equal to 7 root 3, which is what we got. But it might be easier to just memorize that the sides opposite the 60-degree side is going to be the square root 3 times the short side. And the short side is going to be half of the hypotenuse. Anyway, the more practice you do with those, the more it will make sense.
Okay. A square is circumscribed about a circle. What is the ratio of the circle to the area of the square? So the square is circumscribed about the circle. So I’m going to draw the circle and the square. So if that’s my circle, and then if I want to draw a square, see if I can – nope. That’s not what I wanted to do. I wanted to draw a solid square. Let’s me see if I can draw it without pressing the shift key. I think that’s close enough. So we know that the square is on the outside, because it’s about the circle. It’ circumscribed about the circle. What is the ratio of the circle to the area of the square? So let’s say that this is the center of the circle right there, this is its radius. Let’s call that r. Well, what’s the area of the square going to be? If that’s the radius, and this is also the radius, right? So, one side of the square up here is going to be 2r. And so, this side is also going to be 2r. It’s a square. All the sides are the same. So the area of the square – so, they want to know the ratio of the area of the circle to the area of the square. Okay. So area of circle over the area of the square is equal to the area of the square. It’s just 2r times 2r, which is what? 4r squared. That’s 2r times 2r. Area of the circle is just pi r squared. Hopefully, you learned the formula of the area of the circle. It’s just area id pi r squared. So, write that down. Pi r squared. Divide the numerator and denominator by r squared. You’re left with pi to 4. And that’s choice D.
Problem 74. In the circle below, AB and CD are chords intersecting at E. Fair enough. If AE is equal to 5, so this is equal to 5. We pick another color. BE is equal to 12. What is the value of DE? Oh, CE is equal to 6. What is the value of DE? Let’s call that x.
Now, I’m not going to prove that it here just for saving time. But there’s a need property of chords within a circle that if I have two chords intersecting a circle, it turns out the two segments, when you multiply them times each other, are always going to be equal to the same thing. So in this case, 5 times 12, let me write that down. So this 5 and this 12, - so the two segments of chord AB. So 5 times 12, that’s going to be equal to these two segments multiplied by each other, and I’ll do that in a different color, these two segments. It’s going to be equal to x times 6. So you get 60 is equal to 6x. Divide both sides by 6, you get x is equal to 10. And that is choice C. That might be a fun thing for you to think about after this video of why that is. Maybe you want to play around with chords and prove to yourself that that’s always the case. At least that makes intuition for you. Makes sense.
RB is tangent to a circle, right. RB is tangent. Tangent means that it just touches the outside of the circle right there at only one point and it’s actually perpendicular to the radius of that point. So this is the radius of that point, right? The center is at A. This is a radius and it’s tangent to point B, so it’s perpendicular to the radius at that point. And point BD is the diameter. Okay. Fair enough. A is the center, so that’s kind of obvious. Oh, yeah. That’s a line. So that’s obvious. So they want to know, what is the measure of angle CBR? So, they want to know what this angle is equal to. Well, I kind of did it inadvertently. We know that when a line is tangent to a circle, it’s perpendicular to the radius at that point. So this whole angle is 90 degrees. So the angle that we’re trying to figure out, let’s call that x. That’s the compliment to 25, right? X plus 25 is equal to 90. Subtract 25 from both sides. X is equal to 65 degrees. And that is choice B.
Next problem. Oh I’m all out. Okay. I’ll see you in the next video.