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Triangles - Part II
Okay, now factor triangles. In the first triangle video, we talked about ideas that were true for all triangles. In this video, we will begin with two special categories of triangles, isosceles and equilateral triangles. An isosceles triangle is one in which two sides are equal.
Now it happens to be true, if the two sides are equal, then it must be true that the angles opposite those sides are also equal. So those would be these two angles here, they would have to be equal. In fact, Euclid’s remarkable theorem on this topic is a two-way theorem. What do I mean by that? If a triangle has two equal sides, then it must have the opposite angles equal.
And if it has two equal angles, then the opposite sides must be equal. So from the sides, you can deduce the angles, from the angles, you can deduce the sides, it’s a two-way theorem. Could an isosceles triangle have a right angle? Could it have an obtuse angle? You might wanna pause the video and think about this for a moment.
Of course, the answer is yes in both cases. Here are isosceles triangles, one with a right angle and one with an obtuse angle. And of course, we can even find the other angles in that rightang, right triangle. We have 90 degrees so the other two angles also have to add up to 90 degrees. So each one must be 45. In the obtuse triangle, we have 110 degrees.
So the other two sides must, the other two angles must add up to 70 degrees, so that means each one is 35 degrees. An equilateral triangle has three equal sides and three equal angles. Since the three angles have to add up to 180 degrees, each one must equal 60 degrees.
An equilateral triangle always has three 60 degree angles. Here’s an example equilateral triangle. Is an equilateral triangle also an isosceles triangle? Technically yes, technically the definition of isosceles is a triangle with two or more equal sides. That’s the technical definition of isosceles.
In all likelihood, the test will not ask directly about this distinction, but you must know that every special fact about isosceles triangles also applies to equilateral triangles. That’s important to know. Now, we’ll return to all triangles. We talk about area.
The area of a triangle, how much space the triangle takes up? If units are used, this area is measured in square units. Something like square inches or centimeters squared, something of that sort. You may even remember the formula, area equals 1/2 base times height, where b is the base and h is the height.
But these problematic terms need to be clarified. There are some very naive understandings of these, that lead to all kinds of mistakes. So we have to be very clear. Exactly what do we mean by a base and exactly what do we mean by a height. So first of all, what exactly is the base of a triangle?
Naive students will say the bottom side, but this is not the whole answer. Part of the problem is that some textbooks often insist on representing all triangles with a horizontal side on the bottom. This over-representation of one orientation misleads students into mistakenly thinking that every triangle comes with a horizontal bottom side. In fact, if we look at a triangle like this, this is a general triangle, there’s no horizontal bottom side.
The word base has absolutely nothing to do with being horizontal. In fact, in any triangle any of the three sides could be the base. The base is simply the side of the triangle we choose to find the area. So any triangle has three possible bases, those are the three sides. Now, what exactly is the height of a triangle? Just as students naively assume that a base is horizontal, so they naively assume that the height is vertical.
Neither of these is necessarily true. In fact, h is the length of a segment called the altitude. Now the rough definition is an altitude is a segment from a vertex to the opposite side that is perpendicular to that side. So we’ll expand that definition in a moment. Before, for the moment, that, that rough definition is fine.
So just as any side can be the base, we can draw an altitude from any vertex to the opposite side. So here is a triangle with the three altitudes drawn. So notice each one is a different length, so there’s three different heights. And that’s, that’s not a problem. Each height is paired uniquely with a particular base.
So in the first one, we draw the altitude from B. We draw it to base AC. So, whatever side the altitude is hitting, that’s what we’re gonna use as our base, paired with that height. So, one way to calculate the area, would be the length of base AC times altitude BD.
Now, we draw the altitude from A to BC, as we have in the middle triangle. Now we’re choosing BC as the base, because that’s the side that the altitude is hitting. So now the altitude is gonna be 1/2 BC times AE, and similarly on the right we’re drawing the line the altitude from vertex C we’re drawing it to side AB.
So we’re choosing AB as our base and so here the base will be ab, and we’ll have 1/2 AB times CF. It turns out for any triangle, if we calculate these, the area in these three different ways, we’ll always get the exact same number. The triangle has only one area, so these three different ways of calculating the area will always lead to the same answer.
Now, this rough definition of altitude is fine in triangles in which all three angles are acute. Think about an obtuse triangle. There’s no way to draw a segment from P to MN that can be perpendicular to MN. So just impossible. In other words, any segment that we, starts at P and intersects anywhere on segment MN, it’s not gonna be perpendicular to MN.
So, we have a problem. The altitude is not simply a line from a vertex to the opposite side. What we have to do is extend that segment. And the draw, then draw a segment from P that is perpendicular to the extended segment. So notice that the altitude is not actually intersecting segment MN itself.
It’s intersecting the line along which MN lies. So PR is in fact the altitude. It is perpendicular to that line. Technically, it doesn’t intersect the segment MN, it intersects the line on which that segment lies. So we could legitimately say that the area of this triangle is one-half the base, MN, times the altitude, PR.
That is 100% true. Similarly we could rearrange, we could draw altitudes from other base, from other vertices. And we draw one from N, that just intersects the opposite side, that’s easy. We draw one from M. Well again, we have to extend PN the other way.
And then the altitude intersects that extended line. So, for the second triangle with an altitude from N, we can just say that the area is one half MP times SN. And for the third triangle, we can say the area is one half PN times MT. Notice that in right triangles, two of the altitudes would be the, legs, the sides that meet at the right angle.
If one leg is the base, the other leg is the altitude. And either way, the area equals 1/2 times the product of the legs. So, it doesn’t really matter whether we call AC the base and BC the height, or call BC the base and call AC the height, either way we’re gonna wind up with an area 1/2 AC times BC. So, that’s how you find the area of a right triangle.
Sometimes student worry about how to find the length of the altitude. The test will not give you simply the, the three sides of a triangle and expect you to find the altitude. Now technically there would be ways to do this involving trigonometry and other more advanced forms of math.
This is not something you’re expected to do on the test. An altitude always forms a, smaller right triangles within any triangle. That means sometimes we might use the Pythagorean theorem or another right-triangle fact to find the altitude. We will discuss this more in the next video. So yes, there are ways using the Pythagorean theorem, but there’s not a way just with a general triangle that you could automatically compute the altitude.
Here’s a practice problem, pause the video and then we’ll talk about this. Okay, in the diagram two altitudes are drawn and we want the length of the base GF. Well, let’s see. We have the altitude GX which has a length of six and that’s drawn to the line.
The, on which FH lies. So it must be true that we could say that the area is 1/2 6 times 14. If we call GF, if we just say, we’ll leave that as GF because that’s unknown. So that lies along the line GFY. And HY is the altitude to that line. So the area is also 1/2 7 times GF.
So those are two different ways to calculate the area. And here’s the big idea. If we calculate the area of the same triangle in two different ways, those two different ways must lead to the same answer. So we must be able to calculate the area that way. Calculate it this way, and say those two are equal.
Well we’ll divide, get rid of the one half, multiply both sides by two to cancel that. And so we get, we just get this product here. We can plug in the numbers that we have. We’ll divide both sides by seven, and what we’re left with is just 2 times 6 which is 12.
And that must be the length of FG or GF. Now we’re gonna talk about four kinds of lines in a triangle. You will not need to know these names, but you will need to keep these ideas straight. So the first line is the altitude, we just talked about this. This goes to the vertex and it’s perpendicular to the opposite side.
Technically it’s perpendicular to the line that contains the opposite side if the triangle is obtuse, okay? But that definitely goes to the vertex, definitely makes a right angle. Obviously, the point where the altitude intersects the opposite side usually isn’t the midpoint of the opposite side, as we see here. S is not even close to being the midpoint of PR.
The second special line is the perpendicular bisector of a side. Perpendicular bisector side, it goes through the midpoint, it’s perpendicular to the midpoint. But this in general does not even pass through the opposite vertex at all. So that green line, that’s not even pretending to pass through Q. It just goes right by Q without passing it.
So in general, we can construct a perpendicular bisector of a side, but in general that’s not going to pass through the vertex of the opposite side, the opposite vertex. The third special line goes from the vertex to the midpoint of the opposite side, this is called a median. And so here, M is definitely the midpoint of the opposite side.
So we’re drawing the line from Q to M. And in general, the median divides the opposite side in half, but does not divide the angle at Q in half, and in fact if you look at that angle it’s pretty easy to tell that the angle on the left, PQM, that’s a pretty small angle. And the angle on the right, MQR, that’s a much larger angle, in fact that might almost be 90 degrees, or slightly more than 90 degrees.
So clearly, those two angles are not equal. So, this line, divides the opposite side in half, does not divide the opposite angle in half. Does not divide the angle in half at all. The fourth special line is an angle bisector.
This is the opposite. The angle bisector divides the angle in half, but it doesn’t usually divide the opposite side into half. So notice here, where it intersects the opposite side point V, point V is not even pretending to be the mid point of PR. It’s nowhere close to dividing PR in half.
So, we do divide the angle in half. We do not divide the opposite side in half. Very important to keep all those ideas straight. So that’s what’s true in a general triangle. And a general triangle on any side. Any side in opposite angle pair we can construct these fours lines.
The altitude, the perpendicular bisector, the median, and the angle bisector. And they are four different lines doing four different things in a general triangle. You have to appreciate what is or isn’t true in most triangles. In order to appreciate the very special thing that happens only in an isosceles triangle.
The line down the middle of an isosceles triangle, from the vertex to the midpoint of the base, plays all four of these roles at once. So point T is the midpoint of the base. So if we draw the line from G to T, this is gonna be perpendicular to the base, so it’s gonna be an altitude.
Of course, it’s a line to the midpoint of the opposite side, so it’s a median. It also bisects the angle, so it’s an angle bisector. And so it plays all four roles. Median, altitude, angle bi-sector and perpendicular bi-sector. All of four those is played by the single line down the middle of an isosceles triangle.
Now, in general, those four special lines are four different lines in ordinary triangles, so if we get any information that one segment is playing more than one role, that the perpendicular and angle bisectors are the same thing, that’s enough to prove that the triangle is isosceles. So any time any two of those roles are played by the same line right there. That’s an indication the triangle must isosceles, because in general those four different roles are four completely different lines in most triangles.
The midline in an isosceles triangle is a line of symmetry which divide the bigger triangle into two congruent right triangles. Course, all these isosceles triangle facts apply also to the equilateral triangle, which is a special case of isosceles. In summary, isosceles means equal bases and opposite angles are equal.
And in fact it goes both ways. If you know the bases are equal, then you can tr, then you can prove opposite angles are equal. If you know the angles are equal then you can prove the opposite sides are equal. You can go either way. Equilateral has three equal sides and all 60 degree angles.
Area equals 1/2 base times height, but we have to be very careful cause we can’t afford to be naive. Any side can be the base, and the altitude is perpendicular to that side, the length of the altitude is the height. And so, base and height have absolutely nothing to do with being horizontal or vertical.
The altitude, the perpendicular bisector, the line. From the vertex to the opposite midpoint which is the median and the angle bisector are four completely different lines in most triangles. But the line of symmetry in isosceles triangle plays all four of those roles at once. And so that is something very, very special.
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