This blog is the online extension of our intermediate classrooms. Our goal is to enhance and document our learning experience throughout the school year, and share this journey with teachers, parents and students. We welcome your constructive feedback, and we look forward to learning with you!
Grade 7/8 POTW I found multiple ways of this question, all generally having the same general gist. The side of each segment is 2cm with the other lengths being something else, unknown for now. If we take BC and AB and divide the triangle in the same way two more times, now parallel to both the line segments, you'll get a bunch of little equilateral triangles, all having side lengths of 2 cm. There would be a total of 16 of these mini 2 by 2 equilateral triangles. All you have to do is find the area of 1 and multiply by 16. An equilateral triangle of 2 cm length would have a height of 1.732050807568877 cm using some huge formula and multiplying them together and dividing by 2 gives an area of 1.732050807568877 cm squared. Multiplying by 16 gives us an area of 27.7128129211 cm squared. For the second way of solving it, it is pretty much the same thing. First, I take 16 cm and calculate the height. The height is 13.856406460551018 cm. Now we divide that by 8 to get the height of 1 segment. We get 1.73205080757 cm (like before for one little triangle). We then multiply it by 2 to find the height of the trapezoid. It results in 3.46410161514 cm. By adding by 2 cm each time as it is an equilateral triangle, we know that the bases are 6 cm and 10 cm. 6 + 10 = 16 (16 x 3.46410161514)/2 = 27.7128129211 cm squared. There are probably more methods with a similar gist of solving. The area of the shaded trapezoid is 27.7128129211 cm squared.
So we know all of the line segments are 16cm, so AB=16, BC=16cm, and CA=16cm. And since we know there are 8 line segments that divide the triangle that are parellel, we know that the space between the lines are 2cm because 16/8=2.
With this, we can use the formula for area of a trapezoid (b1+b2)/2 * h to figure out the area... of a trapezoid. So, since it's an equilateral triangle, to find out the longer side (b2) you'd just count how many spaces (the space in between the lines) and multiply it by 2. So to find b2, there are 5 spaces, multiply it by 2 (because 2 cm) and we get that b2=10cm. Do the same for b1 and we know that b1=6cm.
So b1=6cm and b2=10cm
Im not sure if I'm taking the proper approach to find the height but here is my "math"
So if we divide line segment CA by 2, we can get 8 and by reversing the pythagorean theorem, we'd be able to find a. (a^2 + b^2 = c^2) Since we know c = 16 and b = 8,
16^2 - 8^2 = a^2 Work through the math and we'll get 13.86cm as the full height of the triangle. Divide that by 8 since there are 8 parts and we'll have the height of each section, multiply it by 2 and we get 3.47cm as the full height. I'll just round it to 3.5cm for simplicity, and we now have our final piece to the formula
b1=6cm b2=10cm h=3.5cm
(6+10)/2 * 3.5 =16/2 * 3.5 =8*3.5 =28cm^2
There for, the trapezoid that's shaded in (hopefully) has an area of 28cm (rounded).
POTW: Info: - Equilateral Triangle split into 8 sections. - Sides are 16cm each - 7 line segments - All line segments are parallel. - What is the area of the shaded trapezoid?
First, I need to know the formula for the area of a trapezoid (which, I'm guessing everyone knows... or else you would have not done so well in measurement). Anyways, the formula is (b1+b2)/2 *h. To solve this question, I need to get the two bases and the height (whoop de doo). To find the bases, we find out the number of spaces from point B to the base we need and multiply by 2 (since they are 2 cm each). Doing this, I get the first base to be 6 cm and the second base to be 10 cm. Now for the height (another whoop de doo). To find the height, I need to find the height of the whole triangle then divide by 8, then multiply by 2. If we use the Pythagorean Theorem (a^2 + b^=c^) we can find the missing value, which in this case is a. Spliting side AC into two parts, we get b to be 8 cm and c to be 16 cm. To find a (you can think of a as the bisector of Angle ABC. DING DING DING VOCAB WORD. I'm sorry, I had to). a^2 + 8^2=16^2. Now woah there. Since 192 isnt a perfect square, we need to do some fancy schmancy stuff called prime factoring. (This isn't really necessary but I want to get an exact answer and not a decimal). We prime factor root192 into 8^2 times 3. Now, if we square root that, the perfect square (8^2) will come out as an 8 and the 3 will remain a root. So our answer for the height will be 8 root3. (Or if you want to, the decimal value is about 13.856. And please don't ask how I know this. Then we divide this by 8 and multiply by 2. 8 root3 over 8 is 1 root3 and that multiplied by 2 is 2 root3. Yay. A height. Now we find the area of the trapezoid.
(6+10)/2 * 2 root3 16/2 *2 root3 8 * 2root3 16 root3. Yay. That's my radical answer. For the decimal answer....
Instead of 8 root3 being the height, the height would then be 3.46 (divded by 8, multiply by 2) (6+10)/2 * 3.46 16/2 * 3.46 8 * 3.46 27.68. Something like that.
I don't really know how to answer this, so I'm gonna try my best. Each side length of the equilateral triangle is 16cm long. It is then divided by 7 line segments to create 8 segments. The space between each line is 2 cm as 16/8 = 2. I need to find the base and top length and height of the trapezoid now to calculate its area. The pattern I found for the lengths of each line segment, is that it increases by 2 each line segment. The first line segment makes an equilateral triangle, which means the length is 2 cm. Then the next cut makes a fit for 3 triangles, but only 2 have a length. This means that the second line segment is 4cm long. I can go up to the 3rd line segment from the left which is 6 cm, the base. Then go 2 more line segments across to find the base length of the trapezoid is 10 cm. The height remains at 2 cm with each line segment as the space between each line segment is 2 cm. A = (b1+b2)/2 * h A = (6+10)/2 * 2 A = 8 * 2 A = 16 cm squared
The area of a trapezoid is h((b1+b2)/2). I know that b1 and b2 are 10 and 6 because the length of the base goes up by 2 as you go down the triangle. I found the height by finding the height of the triangle and then divided it by 4. I got 3.46410161514. A = h((b1+b2)/2) A = 3.46410161514((6+10)/2) A = 3.46410161514(16/2) A = 3.46410161514 * 8 A = 27.7128129211 A = 27.71
The area of the shaded region is 21.71 cm squared.
ReplyDeleteGrade 7/8 POTW
I found multiple ways of this question, all generally having the same general gist.
The side of each segment is 2cm with the other lengths being something else, unknown for now.
If we take BC and AB and divide the triangle in the same way two more times, now parallel to both the line segments, you'll get a bunch of little equilateral triangles, all having side lengths of 2 cm.
There would be a total of 16 of these mini 2 by 2 equilateral triangles. All you have to do is find the area of 1 and multiply by 16.
An equilateral triangle of 2 cm length would have a height of 1.732050807568877 cm using some huge formula and multiplying them together and dividing by 2 gives an area of 1.732050807568877 cm squared. Multiplying by 16 gives us an area of 27.7128129211 cm squared.
For the second way of solving it, it is pretty much the same thing. First, I take 16 cm and calculate the height. The height is 13.856406460551018 cm. Now we divide that by 8 to get the height of 1 segment. We get 1.73205080757 cm (like before for one little triangle). We then multiply it by 2 to find the height of the trapezoid. It results in 3.46410161514 cm. By adding by 2 cm each time as it is an equilateral triangle, we know that the bases are 6 cm and 10 cm.
6 + 10 = 16
(16 x 3.46410161514)/2 = 27.7128129211 cm squared.
There are probably more methods with a similar gist of solving.
The area of the shaded trapezoid is 27.7128129211 cm squared.
POTW:
ReplyDeleteSo we know all of the line segments are 16cm, so AB=16, BC=16cm, and CA=16cm. And since we know there are 8 line segments that divide the triangle that are parellel, we know that the space between the lines are 2cm because 16/8=2.
With this, we can use the formula for area of a trapezoid (b1+b2)/2 * h to figure out the area... of a trapezoid. So, since it's an equilateral triangle, to find out the longer side (b2) you'd just count how many spaces (the space in between the lines) and multiply it by 2. So to find b2, there are 5 spaces, multiply it by 2 (because 2 cm) and we get that b2=10cm. Do the same for b1 and we know that b1=6cm.
So b1=6cm
and b2=10cm
Im not sure if I'm taking the proper approach to find the height but here is my "math"
So if we divide line segment CA by 2, we can get 8 and by reversing the pythagorean theorem, we'd be able to find a. (a^2 + b^2 = c^2) Since we know c = 16 and b = 8,
16^2 - 8^2 = a^2
Work through the math and we'll get 13.86cm as the full height of the triangle. Divide that by 8 since there are 8 parts and we'll have the height of each section, multiply it by 2 and we get 3.47cm as the full height. I'll just round it to 3.5cm for simplicity, and we now have our final piece to the formula
b1=6cm
b2=10cm
h=3.5cm
(6+10)/2 * 3.5
=16/2 * 3.5
=8*3.5
=28cm^2
There for, the trapezoid that's shaded in (hopefully) has an area of 28cm (rounded).
POTW:
ReplyDeleteInfo:
- Equilateral Triangle split into 8 sections.
- Sides are 16cm each
- 7 line segments
- All line segments are parallel.
- What is the area of the shaded trapezoid?
First, I need to know the formula for the area of a trapezoid (which, I'm guessing everyone knows... or else you would have not done so well in measurement). Anyways, the formula is (b1+b2)/2 *h.
To solve this question, I need to get the two bases and the height (whoop de doo). To find the bases, we find out the number of spaces from point B to the base we need and multiply by 2 (since they are 2 cm each). Doing this, I get the first base to be 6 cm and the second base to be 10 cm.
Now for the height (another whoop de doo). To find the height, I need to find the height of the whole triangle then divide by 8, then multiply by 2. If we use the Pythagorean Theorem (a^2 + b^=c^) we can find the missing value, which in this case is a.
Spliting side AC into two parts, we get b to be 8 cm and c to be 16 cm. To find a (you can think of a as the bisector of Angle ABC. DING DING DING VOCAB WORD. I'm sorry, I had to).
a^2 + 8^2=16^2.
Now woah there. Since 192 isnt a perfect square, we need to do some fancy schmancy stuff called prime factoring. (This isn't really necessary but I want to get an exact answer and not a decimal). We prime factor root192 into 8^2 times 3. Now, if we square root that, the perfect square (8^2) will come out as an 8 and the 3 will remain a root. So our answer for the height will be 8 root3. (Or if you want to, the decimal value is about 13.856. And please don't ask how I know this. Then we divide this by 8 and multiply by 2. 8 root3 over 8 is 1 root3 and that multiplied by 2 is 2 root3. Yay. A height. Now we find the area of the trapezoid.
(6+10)/2 * 2 root3
16/2 *2 root3
8 * 2root3
16 root3. Yay.
That's my radical answer. For the decimal answer....
Instead of 8 root3 being the height, the height would then be 3.46 (divded by 8, multiply by 2)
(6+10)/2 * 3.46
16/2 * 3.46
8 * 3.46
27.68.
Something like that.
Mathmatica tomorrow :)
If you don't understand anything here, you can ask me at school. :)
DeleteI don't really know how to answer this, so I'm gonna try my best.
ReplyDeleteEach side length of the equilateral triangle is 16cm long. It is then divided by 7 line segments to create 8 segments. The space between each line is 2 cm as 16/8 = 2.
I need to find the base and top length and height of the trapezoid now to calculate its area. The pattern I found for the lengths of each line segment, is that it increases by 2 each line segment. The first line segment makes an equilateral triangle, which means the length is 2 cm. Then the next cut makes a fit for 3 triangles, but only 2 have a length. This means that the second line segment is 4cm long. I can go up to the 3rd line segment from the left which is 6 cm, the base. Then go 2 more line segments across to find the base length of the trapezoid is 10 cm. The height remains at 2 cm with each line segment as the space between each line segment is 2 cm.
A = (b1+b2)/2 * h
A = (6+10)/2 * 2
A = 8 * 2
A = 16 cm squared
The shaded trapezoid is 28cm^2. (ROUNDED)
ReplyDeleteThe area of a trapezoid is h((b1+b2)/2). I know that b1 and b2 are 10 and 6 because the length of the base goes up by 2 as you go down the triangle. I found the height by finding the height of the triangle and then divided it by 4. I got 3.46410161514.
ReplyDeleteA = h((b1+b2)/2)
A = 3.46410161514((6+10)/2)
A = 3.46410161514(16/2)
A = 3.46410161514 * 8
A = 27.7128129211
A = 27.71
The area of the shaded region is 21.71 cm squared.
POTW:
ReplyDeleteThe area of the shaded region (trapezoid) is 28 cm squared (rounded).
My messy work is done on paper.