6 th Grade Unit 1 Lesson 8 Area of Triangles Illustrative Mathematics

unit one lesson eight area of triangles number one to find the area of this right triangle diego and jada used different strategies diego drew line through the midpoints of the two longer sides which decomposes the triangle into trapezoid and smaller triangle he then rearranged the two shapes into parallelogram explain how diego might use his parallelogram to find the area of the triangle since diego's parallelogram is made out of the same parts as the triangle he can simply multiply the base times the height three times four equals twelve the area of diego's parallelogram is twelve square feet and since the parallelogram is made out of the same parts as the triangle the area of the triangle would also be 12 square feet jada made copy of the triangle rotated it and lined it up against one side of the original triangle so that the two triangles make parallelogram explain how jada might use her parallelogram to find the area of the triangle unlike diego jada didn't make her parallelogram out of the same parts as the triangle since jada used the original triangle and copy of the original triangle the area of her parallelogram is going to be twice the size of the area of the triangle so when we multiply the base times the height 3 times 8 we'll get 24 but since this area is twice the size of the area of the original triangle we'll have to divide that by 2. 24 divided by 2 is 12. the area of the original triangle is 12 square feet number two find the area of the triangle explain or show your reasoning decided to decompose this triangle by removing section and rearranging it so it becomes rectangle now that have rectangle can easily see that the base is six and the height is two six times two is twelve the area of this rectangle is 12 square units and the area of the triangle is 12 square units because the rectangle was made out of the same parts as the triangle i'm going to decompose this triangle and rearrange it so it becomes parallelogram need to make horizontal cut halfway between the base and the opposite vertex and the distance between the base and the opposite vertex is three so i'll have to make horizontal cut one and half units above the base and then rearrange it into the shape of parallelogram the base of the parallelogram is four and the height is one and five-tenths need to multiply the base times the height to get the area so four times one and five-tenths equals six the area of the parallelogram end of the triangle is six square units number three which of the three triangles has the greatest area show your reasoning if you get stuck use what you know about the area of parallelograms to help you i'm going to go ahead and use what know about parallelograms to help me so the first thing that need to do is turn these triangles into parallelograms either by decomposing and then rearranging into the shape of rectangle or copying the shape of the triangle and attaching it to the existing triangle to make rectangle or parallelogram now it's pretty easy for me to find the base and the height of the rectangle and parallelograms but remember the area of the rectangle and parallelograms are twice the area of the original triangles they all have base of 5 units and height of 4 units which is an area of 20 square units so all the triangles would have an area of 10 square units it turns out that neither of the three triangles has the greatest area they all have an equal area of ten square units number four draw an identical copy of each triangle such that the two copies together form parallelogram if you get stuck consider using tracing paper i've drawn identical copies of each triangle and i've joined them together to form parallelograms number five parallelogram has base of three and five-tenths units and corresponding height of two units what is its area again to find the area we need to multiply the base times the height three and five-tenths times two will equal the area three and five-tenths times two is seven so the area would be seven square units parallelogram has base of three units and an area of one and eight tenths square units what is the corresponding height for that base they've told us that the base is 3 units and that the area is 1 and 8 10 square units but they haven't told us the height so we need to figure out the height base times height equals area the base is three the height is unknown and the area is one and eight tenths square units so we need to find out what number times three or three times what number equals one and eight tenths and we can do that by dividing one and eight tenths divided by three will tell us the height one and eight tenths divided by three is six tenths so the height would be six tenths of unit parallelogram has an area of twenty and four tenths square units if the height that corresponds to base is four units what is the base again we know that the base times the height equals the area the base is unknown but we do know that the height is 4 and the area is 20 and 4 10 square units what number times 4 equals 20 and four tenths we can figure that out by dividing twenty and four tenths by four twenty and four tenths divided by four is five and one tenth the base is five and one tenth of unit you