Rectangles Properties of Parallelograms Special Quadrilaterals Geometry

in this video we're going to focus on rectangles we're going to talk about their properties and how to solve problems associated with them now rectangle is quadrilateral and quadrilateral is four-sided polygon now rectangles are also special type of quadrilateral known as parallelograms so let's say if we have this rectangle rectangle abcd like parallelogram opposite sides are parallel so bc is parallel to and you could write it this way in addition to that we could say that is parallel to dc now what else do we know about rectangles just like parallelogram opposite sides are congruent so bc and ad they're congruent so that means that and are also congruent now all angles are right angles they're all equal to 90 degrees so angle angle angle angle they equal 90. now the next thing you need to know are the diagonals the diagonals are congruent to each other so diagonal ac is congruent to diagonal bd now also the diagonals bisect each other so what that means is that ae is congruent to ec and be is congruent to ed so we can write it like this and are congruent and is congruent to which makes the midpoint of and is also the midpoint of dd so those are some basic properties of rectangles now some formulas that you may want to keep in mind are these let's say this is the length and the width of the rectangle the area is the length times width the perimeter is the sum of all four sides so it's 2l plus 2w and if you wish to calculate the length of the diagonal you could use the pythagorean theorem squared is equal to squared plus squared in this case squared is squared plus squared so you can use that to calculate the length of the diagonal now let's work on some example problems our goal is to determine the measure of segment bd and we're given and ec so how can we do so but we know that the diagonals bisect each other so is the midpoint of ac which means that ae and ec they're equal to each other so if we set them equal to each other we could say that squared plus two is equal to three plus six so i'm going to take everything from the right side and move it to the left side so it's going to be squared minus 3x plus 2 minus 6. now 2 minus 6 is negative 4. and so we have trinomial where the leading coefficient is one and we need to factor it in order to solve the quadratic equation so what two numbers multiply to the constant term negative four but add up to the middle coefficient negative three this is going to be negative 4 and positive 1. negative 4 plus 1 adds up to negative 3 and negative 4 times 1 is still negative 4. so to factor it's going to be minus 4 times plus 1. now what we need to do is we need to set each factor equal to zero so minus four is equal to zero and plus one is equal to zero so has to equal four and negative one now let's keep in mind that ae is squared plus two because is squared we won't get negative result for ae for using either of these two answers now for ec is mean is three plus six so if we plug in negative one it will still give us positive result so can be both answers now if is equal to four is going to be four squared plus two which is four squared is sixteen sixteen plus two is eighteen and three times four plus six that's twelve plus six that's also eighteen so ae and ec potentially both 18 which means ac the sum of ae and ec that's going to be 18 plus 18 which is 36 and the diagonals of rectangle are congruent which means that ac and bd they're the same so this diagonal is congruent to that one which means bd is 36 so that's one answer for bd which is what we're looking for now to find the other answer let's use different value so let's start with is still squared plus two so let's replace with negative one negative one squared is one so it's one plus two which gives us three and should give us the same result it's three plus six so that's going to be three times negative one plus six which is negative three plus six and so that's three as well now ac well first let's start with bd bd is equal to ac and ac is the sum of ae plus ec so that's going to be 3 plus 3 which is 6. so we have two potential answers for bd it could equal 36 or 6 in this problem based on the expressions of ae and ec number 2 rectangle abcd has an area of 40 and perimeter of 26. what is the length of segment ae so for those of you who want to try this problem feel free to pause the video and work it out so how can we use the area and the perimeter of this rectangle in order to determine the measure of ae so let's call this how can we determine the value of what do you recommend that we do well let's say that is the length and dc is the width ac is the length of the diagonal we said that the diagonal is equal to well squared is going to be squared plus squared now if we could find which is the length of ac ae is half of that so ae is simply one half of ac and ac is basically the diagonal so if we can calculate we can easily find and in order to find we need to determine the length and the width of the rectangle so how can we do that how can we determine the length and the width given the area and the perimeter well we need to write system of equations the area is the length times the width and the perimeter is 2l plus 2w so we could say that 40 which is the area is lw and the perimeter is 26 that's equal to 2l plus 2w so what we're going to do is we're going to solve by substitution this equation let's isolate so let's divide both sides by so 40 over is equal to and in the second equation let's replace with 40 divided by so 26 is equal to 2l plus 2 times 40 over now to get rid of the fraction let's multiply every term by so this one i'm just going to get rid of this so this is going to equal 26 and that's equal to 2 squared and over here the l's will cancel and we'll just have 2 times 40 which is 80. now let's divide everything by 2. 26 divided by 2 is 13. 2 over 2 is 1 and 80 divided by 2 is 40. now let's take this term move it to the right side so this is going to be 0 is equal to squared minus thirteen plus forty now we need to factor the expression what two numbers multiply to forty but add to negative thirteen five and eight multiplies to forty but we need to use negative 5 and negative 8 because those two numbers add up to negative 13. so to factor it's going to be minus 5 times minus 8. so we have two potential answers could be 5 or could be eight now the area has to be forty so forty is equal to times five times is forty so if is five is eight and when is eight is 5. so regardless if you choose this answer or this answer the length of the diagonal will still be the same and will still be the same so we could just choose one of those two answers so we're going to say that is 8 because it appears to be longer if you look at the shape of this figure and which appears to be the shortest side we're going to say it's 5. so squared is squared plus squared is eight is five eight squared is sixty-four and five times five is twenty-five sixty-four plus twenty-five is eighty-nine so if we take the square root of both sides the diagonal is the square root of eighty-nine ae is one half of the diagonal so ae is half of the square root of 89 so if you want to get decimal value for that this is going to be 4.717 approximately and so this is the exact answer though for ae it's half of the square root of 89. you