POTW #4 –
Challenge Yourself!
Hello again, please remember to try
different problem-solving strategies when stuck on the POTW! Mr. Milette shared
a gapps file with 10 different math problem-solving strategies a few weeks
back.
Please check your last week's
answers against the correct solutions below. The new POTW #4 follows after the
solutions below.
POTW #3 Grade 7 Answer:
POTW #3 Grade 8 Answer:
POTW #4 Grade 7 Question:
POTW #4 Grade 8 Question:
First, I have to find the area of this triangle, and the formula for finding it is base times height, divided by 2.
ReplyDeleteOne of the ways to find area of this triangle is to use the base from A to B and the height of C to D. Base length from A to B is 16 cm, and the height from C to D is 6cm.
We would first have to multiply 16 by 6, and that gives us 96.
After I have to divide 96 by 2 to follow the formula to calculate area, and that gives us 48cm²
Now that we know that the area of the triangle is 48cm², we have to find the length of BE, but we could use the formula to find area.
BE is the height of the triangle, and AC is the base, which is 12cm. Also, when you multiplied BxH previously, the product was 96, and then divided by 2. So, that means you would need to find a number that multiplies with 12 to get 96.
A way you can figure this out is to work backwards. You divide 12 by 96, and that gives you 8, which is the length of BE.
To double check, you could look back at the formula and do it again. So...
12x8=96
96/2=48cm²
The length of BE is 8cm.
I'm also not too sure if this is the correct answer, but I tried to answer it as best as I can.
Grade 7 POTW:
ReplyDeleteFirst of all, I had to figure out the area of the triangle. I already know the base, AB, which is 16, and the height, CD, which is 6. So, using the triangle area formula, I get the equation 16*6/2.
16*6 is 96, and dividing it by 2 is 48. The area of the triangle is 48.
Looking at the triangle more closely, I can see that BE can also be a height since it is perpendicular to AC. AC would be the base, and BE would be the height. I already know that the length of AC is 12. Using the triangle area formula again but differently, I calculate backwards using opposite operations and get the equation 48*2/12, which will get me the height.
48*2 is 96, and dividing it by 12 will get me 8.
The length of line segment BE is 8 centimeters.
Alan
Grade 8 POTW:
ReplyDeleteI found the grade 8 POTW to be very complex, though I actually managed to figure out an answer(I have completely no idea if it is right or not :p).
First of all, I have to figure out DM and DC, which are the same because AM is a median, as explained in the question.
Using the Pythagorean Theorem, I can figure out that DM and DC would equal to 5, since 13 squared subtracted by 12 squared would give me 25. I square root this and get 5. (This is still the right formula, but a bit different.)
Now that I know DC is 10, I use some algebra. So let's say X is the length of DB. The length AB would equal to AD added to X, which is 12+X, AC would equal to the square root of 244(Again using the Pythagorean Theorem, since DC is 10, AD is 12, both squared, added together, and the sum being square rooted is the square root of 244), and BC having a length of 10 squared plus X squared and square rooted. So, using the three values I again use the Pythagorean theorem to try to figure out X.
The equation would be: 10 squared + X squared, square rooted and also squared(According to the formula) + 244 square rooted and squared, = (12+X) squared.
Now, I simplify it to get my answer.
I need to change the right side of the equation, using a formula like this:
(a+b) squared = a squared + b squared + 2ab(2*a*b).
So, (12+x) squared would be equal to 144 + X squared + 2*12*X.
Now that both sides are simplified, I can see that there is an X squared on both sides of the equation. I can either cancel it out, or subtract it from both sides.
This leaves me with a final, and very simple equation: 100+244=144+24X . (If you write it out it will be easier to understand.)
To separate X, I have to subtract 144 for both sides.
The left side would be equal to 200, and the right side will be equal to 24X.
To separate X by itself, I would have to divide both sides by 24.
The left side will be equal to 8 and 1/3, while the right side will be X.
The length of BD is 8 and 1/3.
Alan
This is how I went about solving the Grade 7 POTW:
ReplyDeleteThe formula to determine the area of a triangle is (base * height) / 2. Our first step would be to do line CD (the height) multiplied by line AB (the length) divided by 2. In other words, we would be doing (16 * 6) / 2. After doing calculations, we find out that the answer to that expression would be 48 cm^2. That would be the area of triangle ABC. But the objective of this question is to find out the length of line BE. To do this, we simply add line BE to the equation. We would do line BE multiplied by line AC divided by 2. They have also told us that line AC equals 12. Now we start including the area of triangle ABC, which we now know is 48. Whatever equation we follow this up with, the answer would have to equal 48. So, we do (BE * 12) / 2. To make this easier, we could answer the parts that we already know. So, we use the 6 given by the 12 divided by 2 part of the expression. The equation would now become 48 = BE * 6. To find out what BE is, we could use the opposite operation on the two numbers given. In other words, the expression now looks like this: BE = 48 / 6. Once we do 48 divided by 6, we find that the quotient would be 8. Therefore, the length of line BE is equal to 8.
Nice explanation! We came to the same answer however, yours is better explained! Nice job!
DeletePOTW GRADE 7: This is not my strong suit but knowing that the longest side is 16 and the formula for the area is base X height divided by 2, I would guess it is 8?
ReplyDelete12 x 8 = 96
ReplyDelete96 divided by 2 = 48
so, ABC = 48
BE X 12 divided by 2= 48, so BE=8
The answer is 8
In order to solve this problem, we first must figure out the area of the triangle. CD is the height, and AB is the length of the triangle. The formula to find the area of a triangle is Height x Length/2. Since CD is 6, and AB is 16, we can plug the information into the formula.
ReplyDelete(16x6)/2=
96/2=48
Therefore, the area of the triangle ABC is 48cm2. Now since we know the area, we can figure our the length of BE. Since we already figured out the area, we can plug BE and AC into the equation (since they can act as height/length, and see what the line length is (since we know the area and can work backwards).
(BExAC)/2=48
(BEx12)/2=48
(BEx12)=48x2
BEx12=96
BE=96/12
BE=8
Therefore, the length of line BE is 8.
if only they told us that /\ BDC was an equilateral triangle
ReplyDelete̅
T0 solve this question I based my answer off of using the Pythagorean theorem. First I solved DM by using the equation of 13^2-12^2=B^2. Thus I got DM=5. Then after that I multiplied 5 by 2 to get 10 to find DC because M was a median. After that I solved for AC to get 15.06. Then after that I used a formula that involves similar triangles. Thus I got AD over AC is equivalent to DC over CB. Thus after plugging the numbers in I got CB= 7.9. Finally to get DB I used the Pythagorean theorem again also rounding up 7.9 to 8. Thus finally I got 12.8 rounding up to 13.
ReplyDeleteFor the Grade 7 POTW, you first want to find the area, with the formula (bxh)/2. We know that the base is AB (16cm) and the height is CD (6cm). So 16*6 is 96, and divide that by two equals 48. So the area is 48cm2. To find the length of BE, you just invert the formula. Instead of (b*h)/2, it is 2*a/b (A=area). So (48*2)/2 is 8. So the length of BE is 8 cm.
ReplyDeleteSo what I'm gonna do is work forwards then backwards but with different variables. So first I'm gonna find the area using AB as the base and CD as the height. AB = 16 and CD = 6. 16 x 6 = 96, we don't need to know the area for the next step. Now the third known number, AC = 12. AC can be the base if we turn the triangle so now our only missing number is the height of the triangle if AC is the base which is BE. Now we can divide 96 by 12 which is equal to 8 and if 8 was the height, 12 x 8 = 96. This is why BE = 8 cm.
ReplyDeleteThe length of BE is 8 cm. You can get this by first calculating the area of the triangle. the formula is base times height divided by 2. 16*6 is 96. Divided by 2 is then 48. Since the area doesn't change, to find the height of BE the new formula is area/base*2. 48 divided by 12 (because the base is flipped if we're calculating BE) times 2 is 8 so then the length of BE is 8 cm.
ReplyDeleteGrade 7 POTW:
ReplyDeleteFirst off we need to find the area of this triangle. The formula would be base x height/2=area. That would equal 16 x 6=96/2=48cm2 as the area.
Now we can figure out the BE by using it as a height because it looks sideways to that of the one we used earlier. Now we will reverse the area equation to find out the length.
Since we know the length is 12 we can reverse it.
48x2/12=8cm2
That means that the answer is 8cm2
~Michelle
I have to find the area first. The formula is height x base/2. The height is CD and the base is AB.
ReplyDelete6 cm x 16 cm/2 = 96 cm/2 = 48 cm2.
The area of the triangle is 48 cm2. The area is the same if I use AC as the base. It would be 12 cm(AC) x EB(unknown)/2 = 48cm2.
That is the same as (48 x 2)/12.
(48 x 2)/12 = 96/12 = 8.
The missing side(EB) is 8 cm long.
BE is 8 cm long.
I did the grade 8 Potw hard copy and I got 8.3 repeating
ReplyDeleteGrade 8 POTW: Here is how I solved it:
ReplyDeleteFirst I wrote down everything I knew already, that side AD was 12 cm and AM is 13 cm, next I wrote all the right angles. Then I used Pythagorean theorem to solve the problem, I first used it to find out how long side DM was:
12^2 + DM^2 = 13^2
13^2 - 12^2 = DM^2
DM^2 = 25
DM = 5
Then since AM cuts directly into the middle of CD, that would mean the other side ( MC) is also 5 cm, making CD 10 cm in total. Since DC is the attitude, as long as I found out how long it was, I was able to find out how long DB is. Using the Attitude rule, I found out how long DB was:
AD/CD = CD/ DB
12/10 = 10/DB
1.2 = 10/ DB
1.2DB = 10
10 / 1.2 = DB
DB = 8.33333( repeating)
Side length DB is 8.333( repeating) cm long or if your round down, 8 cm long.
I used the Pythagorean Theorem for this. I got 25/3 for the length of BD.
ReplyDeleteAD2+DM2= AM2
144+DM^2=169
DM = square root of 25 = 5 cm
I solved for some of the values and got DM = 5 Meaning CD is 10.
DC is the altitude, so we need to find out how long it is using the altitude rule.
This written as
CD is altitude, so:
AD/CD = CD/DB
12/10 (1.2) = 10/BD
1.2 BD = 10
BD = 10/1.2
10 divided by 1.2 is 8.333, or 25/3 (8 1/3)
The side length of BD is 8.333
GRADE 7 POTW
ReplyDeleteSo first, I have to find out the area of the triangle, ABC and the formula: Base x Height/2 = A
16cm*6cm/2=48cm2
We know now the area is 48 so we can use BE as the height in the equation, reverse it and we got how long BE is!
(48 x 2)/12 = 8.
So the length of BE equals to 8!
First, as many other people have explained, you need to find the area of the triangle. Since we don't know what B E is, we can use A B as the base and C D as the height. Then we do the formula (base x height)/2 to find the area (16x6=96=48). Well now that we have figured out the BxH, we can reverse the formula to find out the length of B E. We know that the product is 96 from the previous calculation, and that A B is 12 cm. So now we have to find a number the can be multiplied with 12 to get 96. So an easy way to do that is divide 96 by 12 to get an answer of 8. So the length of B E should in theory be 8cm.
ReplyDelete(P.S.) I didn't really understand how i was supposed to do this but when looking at some of my peers (mainly max) i sort of grasped an idea, so thanks :D
12 x 8 = 96, 96 divided by 2 = 48, ABC = 48
ReplyDelete96/12 = 8
This means that BE is 8.
The length of BE is 8, work hardcopy
ReplyDelete8, hardcopy
ReplyDelete