Understanding Special Quadrilaterals

In this video, we're going to talk about quadrilaterals. So what exactly is a quadrilateral? A quadrilateral is simply a four-sided polygon. So this would be considered a quadrilateral. It has four sides. Now, there are some special types of quadrilaterals. The first one we're going to talk about is a square. Now, a square has four congruent sides. So each of these sides are congruent to each other. Now, it also has four right angles. All of these are 90 degrees. They're all equal to each other. And so that's one special type of a quadrilateral. Now, let's say the square has a side length S. The area of a square is simply left times width or s times s, which you can write as s squared. The perimeter is the sum of the entire length around the square. So if you add s four times, you get 4s. So here's the practice problem for you. Let's say if you have a square with an area of 36 square inches. What's the side length of the square, and what's the perimeter? Feel free to pause the video. So we know that the area is S squared, so 36 is S squared. If we take the square root of both sides, the square root of 36 is 6. So that means that the side length of the square is 6 units. Now the perimeter is simply 4s. So that's going to be 4 times 6, so the perimeter is 24 inches. And that's how you can calculate the perimeter and the side length given the area of a square. Now the next special type of quadrilateral that we're going to talk about is the rectangle. So what are some features of rectangles? Like a square, all of the angles are congruent. They're all right angles. They all have a measure of 90 degrees. Unlike a square, not all four sides are congruent. Opposite sides are congruent to each other. And so that's a rectangle. Now let's say this is the length and this is the width. The area of a rectangle is length times width. And the perimeter, this is also L, this is W, so the perimeter is 2L plus 2W. So here's a question for you. Let's say if we have this particular rectangle. It has a width of 5 and a length of 12. Calculate the area of the rectangle, also the perimeter, and also the length of the diagonal. So this is the diagonal, which we'll call D. Go ahead and find those things. So first, let's calculate the area. It's length times width, so it's 12 times 5, therefore it's 60 square units. So if this was 5 feet, and this is 12 feet, then it's 60 square feet. The perimeter is going to be 2L plus 2W. So that's 2 times 12 plus 2 times 5. So this is 24 plus 10, which is 34 feet. Now what about the left of the diagonal? How can we find that answer? So notice that we're looking for the hypotenuse of a right triangle. So whenever you have a right triangle, you could use this equation. This is a, b, and c. So you could use the Pythagorean theorem. c squared is equal to a squared. In this example, C is the length of the diagonal, A is 12, and B is 5. 12 squared is 144, and 5 times 5 is 25. 144 plus 25, that's 169. Now, we need to take the square root of 169, which is 13. So the length of the diagonal is 13 feet. And so that's it for this problem. Now let's talk about the next special type of quadrilateral that we need to be familiar with. And this one is the rhombus. So what are some features of the rhombus? What do we need to know? Like a square, all four sides are congruent. Now, unlike a square, not all four angles are congruent. However, opposite angles are congruent to each other. So these two angles are the same, and those two angles are the same. So, for instance, let's... Let's say if this is 100 degrees, then the opposite angle is also 100. Now what do you think this angle is? For a rhombus, consecutive angles must add to 180, they're supplementary. So 180 minus 100 is 80, therefore x has to be 80. And this is also 80. So as you can see, opposite angles are congruent. And consecutive angles are supplementary. Now, what else do we need to know about a rhombus? What are some other features? For example, what if we want to calculate the area of a rhombus? How can we do so? Let me draw a better picture. Now, you need to know about the diagonals of a rhombus. We'll call this D1. And then this is the second diagonal, D2. Let's say D1 is, let me come up with a good number. Let's say this is 12. and let's say D2 is 16 units. With this information can you calculate the area of this particular rhombus? The area of a rhombus is one half times the product of the diagonal, so D1 times D2. So in this example, it's going to be half of 12 times 16. Half of 12 is 6, and 6 times 16, that's 96 square units. So that's the area of this particular rhombus. Now what about calculating the perimeter of a rhombus? So given these two diagonals, 12 and 16, go ahead and pause the video. Try to calculate the diagonal, I mean the perimeter around the rhombus. So you've got to find the length of these four sides and add them up. Now you need to know that the diagonals of a rhombus, first, they form right angles. They're perpendicular to each other. So this is 90 degrees, this is 90, that's 90, and that's 90. Now the second thing you need to know is that these two diagonals, they bisect each other. So diagonal 1 is 12. That means it's going to be split into two equal parts, 6 and 6. Diagonal 2 is 16 units long. If we divide... divide that by 2, you get 8. Now notice that we have 4 right triangles. So this is a right triangle, which means we can calculate the hypotenuse of the right triangle. So let's call the hypotenuse C. That means this is A and this is B. So we can use this equation C squared is equal to A squared plus B squared. But let me make some more space, so I'm just going to get rid of this. so we're looking for the value of C a is six and B is eight so six squared is 36 eight times a is 64 and 36 plus 64 is 100 now if we take the square root of both sides, the square root of 100 is 10. So we can see we have a 6, 8, 10 triangle. And this is also a 6, 8, 10 triangle. And as we said before, all four sides of a rhombus is, they're all congruent to each other. So therefore, the perimeter is going to be 4 times 10, which is 40 units long. So that's how you can calculate the perimeter of a rhombus. Now, before we move on to the next type of quadrilateral, there's one more thing that I want to mention in regard to a rhombus. Now let's go back to previous example that we had what we said this angle was 100 and This is 100 and this is 80 and this is 80 What you need to know is that the diagonals they also bisect the angles So if this is 100, that means this is going to be 50 and this side is going to be 50. Now we still have a right triangle. So the diagonals they still form right angles now this we bisected into two congruent angles So this will be 40 and this will be 40 and as we can see each triangle Will have an interior angle of 180 so 40 plus 50 plus 90 is 180 So that means this is going to be 40 as well and on this side. It's 50 And so that's something else that you want to keep in mind in regards to a rhombus. Now let's move on to the next type of quadrilateral that we need to be familiar with. And that is the kite. So what are some characteristics of a kite that you know of? Let's call this A, B, C, and D. AB is congruent to AD, and BC and DC, they're congruent to each other. So not all four sides are congruent, just some sides are. Now, if we draw two diagonals, let's call this D1. And if we draw another diagonal, which we'll call D2. Now, a kite and a rhombus, they're similar in some ways. These two diagonals, they are perpendicular to each other. So that's one thing. And if we call this point E, you need to know that be an ED they're congruent to each other so AC is the perpendicular bisector of BD however notice that a e and e c are not congruent. This side is clearly longer than that side. So they're not both bisectors of each other, but only one of the diagonals bisects the other one. So just keep that in mind. Now to calculate the area of a kite, it's very similar to the area of a rhombus. It's one half d1 times d2. So let's say if BE and ED are 12. Let's say EC is 5 and AE is 16. What is the area of this particular kite? So notice that D1 is 16 plus 5. So therefore D1 is 21 units long. D2 is 12 plus 12, which is 24 units long. So the area is going to be 1 half D1, which is 21, times D2, which is 24. Half of 24 is 12, so this becomes 12 times 21. So 12 times 21 is 252 square units. So that's the area of this particular kite. Now let's calculate the perimeter of a kite. So like a rhombus, we need to use the Pythagorean theorem to calculate the hypotenuse of the triangles. So let's focus on the 5-12 right triangle. So this is 5, this is 12. We need to calculate the hypotenuse. So c squared is equal to a squared plus b squared. In this case, a will be 5 and b will be 12. 5 times 5 is 25. 12 squared is... 144 and so this adds up to 169 and the square root of 169 is 13 so this is 13 and the other side which is congruent to it is also 13 Now let's find the length of AB, which is the same as the length of AD. So this time, let's make A 16 and B is going to be 12. So we have a 12-16 right triangle. 16 times 16 is 256, and 12 times 12 is 144. 256 plus 144 is 400. And the square root of 400 is 20. So this side is 20, and that side is 20. So the perimeter is simply the sum of all four sides. So it's 20 plus 20 plus 13 plus 13. 20 plus 20 is 40. 13 plus 13 is 26. 40 plus 26 is 66. And so that's the perimeter of this particular kite. It's 66 units. Now there's one more thing that I want to mention in regards to the kite. And that is that angle B and angle D are congruent to each other. So only one pair of opposite angles are congruent. Now the next thing is that AC bisects angle BAD. That means that... These two angles are congruent. Also, AC bisects angle BCD, which means that angle BCE is congruent to angle DCE. So those are some of the things that you want to keep in mind in regards to a kite. Now let's move on to our next shape, and that is a parallelogram. So what do you know about a parallelogram? What are some features that come to mind? For a parallelogram, opposite sides are parallel. So first, let's call this A, B, C, and D. So BC is parallel to AD. And also... AB is parallel to DC. Now, this is true of a square and even of a rectangle. It's also true for a rhombus, opposite sides are parallel. Now, there's some other features that you need to know in regards to a parallelogram. and that is that opposite sides are congruent. So BC and AD are congruent, AB and CD are congruent to each other. Now the next feature are the angles. Opposite angles are congruent. So angle B and angle D are congruent to each other, and A and C are congruent. Now, if angle B and angle D are both 110, what's A and C? Let's call it X. What is the value of X in this example? There's two ways to find the answer. The first method is... that all four angles of a quadrilateral must add to 360. So angle A plus angle B plus angle C plus angle D, these four angles must add to 360. So A is X, B is 110, C is X, and D is 110. X plus X is 2X, so 110 plus 110 is 210. And 360 minus 220 is 140. And if we divide by 2, 140 divided by 2 is 70. Now, something else that you need to know, which could help us to find the same answer a lot faster, is that consecutive angles are supplementary. So, A and B adds up to 180. B and C adds up to 180. C and D adds up to 180. So, consecutive angles are supplementary. Now there's one more thing we need to talk about in regards to parallelograms, and that is the area. So let's say if the base is 10, and the height of the parallelogram is 8. What's the area? The area of a parallelogram is the base times the height. So in this example, it's going to be 10 times 8, which is 80. So this is the height of the parallelogram. Now let's move on to a trapezoid. So what are some things that we need to know about a trapezoid? This is called B1 and B2. Notice that the bases are parallel to each other. And this is the height of the trapezoid. The area of a trapezoid is 1 half B1 plus B2 times the height. So let's say if B1 is 10 and base 2 is, let's say, 20. And let's say the height is 12. So the area of a trapezoid is basically the average baseline times the height. To find the average, you need to add up two things and divide by two. So, the upper base is 10, the lower base is 20, and the height is 12. What's the average of 10 and 20? 10 plus 20 is 30, half of 30 is 15. So the average base length, which will be a line somewhere in the middle, is 15 units long. So it becomes 15 times 12, which is 180. So that's the area of this particular trapezoid. Now this is another type of trapezoid that you need to be familiar with. And it's called an isosceles trapezoid. It's isosceles if these two sides are congruent. And if those sides are congruent, then the base angles are congruent as well. And then these two will be parallel to each other. So that's an isosceles trapezoid. And so that's all I have for this video. Hopefully it gave you a good understanding of the different types of special quadrilaterals out there. And the properties of those quadrilaterals. So thanks again for watching.

In this video, we're going to talk about quadrilaterals. So what exactly is a quadrilateral? A quadrilateral is simply a four-sided polygon. So this would be considered a quadrilateral.

It has four sides. Now, there are some special types of quadrilaterals. The first one we're going to talk about is a square. Now, a square has four congruent sides. So each of these sides are congruent to each other.

Now, it also has four right angles. All of these are 90 degrees. They're all equal to each other.

And so that's one special type of a quadrilateral. Now, let's say the square has a side length S. The area of a square is simply left times width or s times s, which you can write as s squared.

The perimeter is the sum of the entire length around the square. So if you add s four times, you get 4s. So here's the practice problem for you.

Let's say if you have a square with an area of 36 square inches. What's the side length of the square, and what's the perimeter? Feel free to pause the video. So we know that the area is S squared, so 36 is S squared.

If we take the square root of both sides, the square root of 36 is 6. So that means that the side length of the square is 6 units. Now the perimeter is simply 4s. So that's going to be 4 times 6, so the perimeter is 24 inches. And that's how you can calculate the perimeter and the side length given the area of a square. Now the next special type of quadrilateral that we're going to talk about is the rectangle.

So what are some features of rectangles? Like a square, all of the angles are congruent. They're all right angles.

They all have a measure of 90 degrees. Unlike a square, not all four sides are congruent. Opposite sides are congruent to each other.

And so that's a rectangle. Now let's say this is the length and this is the width. The area of a rectangle is length times width. And the perimeter, this is also L, this is W, so the perimeter is 2L plus 2W. So here's a question for you.

Let's say if we have this particular rectangle. It has a width of 5 and a length of 12. Calculate the area of the rectangle, also the perimeter, and also the length of the diagonal. So this is the diagonal, which we'll call D.

Go ahead and find those things. So first, let's calculate the area. It's length times width, so it's 12 times 5, therefore it's 60 square units. So if this was 5 feet, and this is 12 feet, then it's 60 square feet.

The perimeter is going to be 2L plus 2W. So that's 2 times 12 plus 2 times 5. So this is 24 plus 10, which is 34 feet. Now what about the left of the diagonal? How can we find that answer? So notice that we're looking for the hypotenuse of a right triangle.

So whenever you have a right triangle, you could use this equation. This is a, b, and c. So you could use the Pythagorean theorem. c squared is equal to a squared.

In this example, C is the length of the diagonal, A is 12, and B is 5. 12 squared is 144, and 5 times 5 is 25. 144 plus 25, that's 169. Now, we need to take the square root of 169, which is 13. So the length of the diagonal is 13 feet. And so that's it for this problem. Now let's talk about the next special type of quadrilateral that we need to be familiar with. And this one is the rhombus. So what are some features of the rhombus?

What do we need to know? Like a square, all four sides are congruent. Now, unlike a square, not all four angles are congruent.

However, opposite angles are congruent to each other. So these two angles are the same, and those two angles are the same. So, for instance, let's... Let's say if this is 100 degrees, then the opposite angle is also 100. Now what do you think this angle is?

For a rhombus, consecutive angles must add to 180, they're supplementary. So 180 minus 100 is 80, therefore x has to be 80. And this is also 80. So as you can see, opposite angles are congruent. And consecutive angles are supplementary. Now, what else do we need to know about a rhombus?

What are some other features? For example, what if we want to calculate the area of a rhombus? How can we do so? Let me draw a better picture.

Now, you need to know about the diagonals of a rhombus. We'll call this D1. And then this is the second diagonal, D2. Let's say D1 is, let me come up with a good number. Let's say this is 12. and let's say D2 is 16 units.

With this information can you calculate the area of this particular rhombus? The area of a rhombus is one half times the product of the diagonal, so D1 times D2. So in this example, it's going to be half of 12 times 16. Half of 12 is 6, and 6 times 16, that's 96 square units. So that's the area of this particular rhombus. Now what about calculating the perimeter of a rhombus?

So given these two diagonals, 12 and 16, go ahead and pause the video. Try to calculate the diagonal, I mean the perimeter around the rhombus. So you've got to find the length of these four sides and add them up. Now you need to know that the diagonals of a rhombus, first, they form right angles.

They're perpendicular to each other. So this is 90 degrees, this is 90, that's 90, and that's 90. Now the second thing you need to know is that these two diagonals, they bisect each other. So diagonal 1 is 12. That means it's going to be split into two equal parts, 6 and 6. Diagonal 2 is 16 units long.

If we divide... divide that by 2, you get 8. Now notice that we have 4 right triangles. So this is a right triangle, which means we can calculate the hypotenuse of the right triangle. So let's call the hypotenuse C. That means this is A and this is B.

So we can use this equation C squared is equal to A squared plus B squared. But let me make some more space, so I'm just going to get rid of this. so we're looking for the value of C a is six and B is eight so six squared is 36 eight times a is 64 and 36 plus 64 is 100 now if we take the square root of both sides, the square root of 100 is 10. So we can see we have a 6, 8, 10 triangle. And this is also a 6, 8, 10 triangle. And as we said before, all four sides of a rhombus is, they're all congruent to each other.

So therefore, the perimeter is going to be 4 times 10, which is 40 units long. So that's how you can calculate the perimeter of a rhombus. Now, before we move on to the next type of quadrilateral, there's one more thing that I want to mention in regard to a rhombus. Now let's go back to previous example that we had what we said this angle was 100 and This is 100 and this is 80 and this is 80 What you need to know is that the diagonals they also bisect the angles So if this is 100, that means this is going to be 50 and this side is going to be 50. Now we still have a right triangle.

So the diagonals they still form right angles now this we bisected into two congruent angles So this will be 40 and this will be 40 and as we can see each triangle Will have an interior angle of 180 so 40 plus 50 plus 90 is 180 So that means this is going to be 40 as well and on this side. It's 50 And so that's something else that you want to keep in mind in regards to a rhombus. Now let's move on to the next type of quadrilateral that we need to be familiar with.

And that is the kite. So what are some characteristics of a kite that you know of? Let's call this A, B, C, and D.

AB is congruent to AD, and BC and DC, they're congruent to each other. So not all four sides are congruent, just some sides are. Now, if we draw two diagonals, let's call this D1. And if we draw another diagonal, which we'll call D2.

Now, a kite and a rhombus, they're similar in some ways. These two diagonals, they are perpendicular to each other. So that's one thing. And if we call this point E, you need to know that be an ED they're congruent to each other so AC is the perpendicular bisector of BD however notice that a e and e c are not congruent.

This side is clearly longer than that side. So they're not both bisectors of each other, but only one of the diagonals bisects the other one. So just keep that in mind.

Now to calculate the area of a kite, it's very similar to the area of a rhombus. It's one half d1 times d2. So let's say if BE and ED are 12. Let's say EC is 5 and AE is 16. What is the area of this particular kite? So notice that D1 is 16 plus 5. So therefore D1 is 21 units long.

D2 is 12 plus 12, which is 24 units long. So the area is going to be 1 half D1, which is 21, times D2, which is 24. Half of 24 is 12, so this becomes 12 times 21. So 12 times 21 is 252 square units. So that's the area of this particular kite. Now let's calculate the perimeter of a kite.

So like a rhombus, we need to use the Pythagorean theorem to calculate the hypotenuse of the triangles. So let's focus on the 5-12 right triangle. So this is 5, this is 12. We need to calculate the hypotenuse. So c squared is equal to a squared plus b squared. In this case, a will be 5 and b will be 12. 5 times 5 is 25. 12 squared is...

144 and so this adds up to 169 and the square root of 169 is 13 so this is 13 and the other side which is congruent to it is also 13 Now let's find the length of AB, which is the same as the length of AD. So this time, let's make A 16 and B is going to be 12. So we have a 12-16 right triangle. 16 times 16 is 256, and 12 times 12 is 144. 256 plus 144 is 400. And the square root of 400 is 20. So this side is 20, and that side is 20. So the perimeter is simply the sum of all four sides.

So it's 20 plus 20 plus 13 plus 13. 20 plus 20 is 40. 13 plus 13 is 26. 40 plus 26 is 66. And so that's the perimeter of this particular kite. It's 66 units. Now there's one more thing that I want to mention in regards to the kite.

And that is that angle B and angle D are congruent to each other. So only one pair of opposite angles are congruent. Now the next thing is that AC bisects angle BAD.

That means that... These two angles are congruent. Also, AC bisects angle BCD, which means that angle BCE is congruent to angle DCE.

So those are some of the things that you want to keep in mind in regards to a kite. Now let's move on to our next shape, and that is a parallelogram. So what do you know about a parallelogram? What are some features that come to mind? For a parallelogram, opposite sides are parallel.

So first, let's call this A, B, C, and D. So BC is parallel to AD. And also... AB is parallel to DC.

Now, this is true of a square and even of a rectangle. It's also true for a rhombus, opposite sides are parallel. Now, there's some other features that you need to know in regards to a parallelogram. and that is that opposite sides are congruent.

So BC and AD are congruent, AB and CD are congruent to each other. Now the next feature are the angles. Opposite angles are congruent. So angle B and angle D are congruent to each other, and A and C are congruent. Now, if angle B and angle D are both 110, what's A and C?

Let's call it X. What is the value of X in this example? There's two ways to find the answer. The first method is... that all four angles of a quadrilateral must add to 360. So angle A plus angle B plus angle C plus angle D, these four angles must add to 360. So A is X, B is 110, C is X, and D is 110. X plus X is 2X, so 110 plus 110 is 210. And 360 minus 220 is 140. And if we divide by 2, 140 divided by 2 is 70. Now, something else that you need to know, which could help us to find the same answer a lot faster, is that consecutive angles are supplementary.

So, A and B adds up to 180. B and C adds up to 180. C and D adds up to 180. So, consecutive angles are supplementary. Now there's one more thing we need to talk about in regards to parallelograms, and that is the area. So let's say if the base is 10, and the height of the parallelogram is 8. What's the area?

The area of a parallelogram is the base times the height. So in this example, it's going to be 10 times 8, which is 80. So this is the height of the parallelogram. Now let's move on to a trapezoid. So what are some things that we need to know about a trapezoid? This is called B1 and B2.

Notice that the bases are parallel to each other. And this is the height of the trapezoid. The area of a trapezoid is 1 half B1 plus B2 times the height.

So let's say if B1 is 10 and base 2 is, let's say, 20. And let's say the height is 12. So the area of a trapezoid is basically the average baseline times the height. To find the average, you need to add up two things and divide by two. So, the upper base is 10, the lower base is 20, and the height is 12. What's the average of 10 and 20?

10 plus 20 is 30, half of 30 is 15. So the average base length, which will be a line somewhere in the middle, is 15 units long. So it becomes 15 times 12, which is 180. So that's the area of this particular trapezoid. Now this is another type of trapezoid that you need to be familiar with. And it's called an isosceles trapezoid.

It's isosceles if these two sides are congruent. And if those sides are congruent, then the base angles are congruent as well. And then these two will be parallel to each other.

So that's an isosceles trapezoid. And so that's all I have for this video. Hopefully it gave you a good understanding of the different types of special quadrilaterals out there. And the properties of those quadrilaterals. So thanks again for watching.

Transcript for:Understanding Special Quadrilaterals

Transcript for:
Understanding Special Quadrilaterals