finding the area of parallelogram / combination/ permutation

[jigoku_snow]

1) ABCD is a parallelogram and the coordinates of A, B, C are (-8, -11), (1, 2) and (4, 3) respectively. Find the area of the parallelogram.

[i]i manage to find that D =( -5.-10) . area of parallelogram = AD x height i'm stuck here. how to find the height? i tried it by finding the angle between AB and AD first , (37.32 degree) then use it to find the vertical line from B. but i dont get the right answer.

2). Points A and C have coordinates (-1, 2) and (9, 7) respectively. A rectangle ABCD has AC as a diagonal. Calculate the possible coordinates of B and D if the length of AB is 10 units.

from the question, i know that AB=CD=10. the equation of line AB = x^2 +y^2+ 2x -4y-97=0. the equation of line CD= a^2 +b^2-18 a- 14b +30=0. then what should i do next?

3)If the five letters a, b, c, d, e are put into a hat, in how many ways could you draw two letters?

must we use permutation or combination? if permutation, why? why can't i use combination? 5C2?

4))3 balls are to be placed in 3 different boxes, not necessarily with 1 ball in each box. any box can hold up to 3 balls. find the number of ways the balls can placed if
i) they are all of the same colour
ii) they are all of different colours

i) what is wrong with my workings? 3C3 + (3C2 x 1C1) + 3C1?
ii) i have no clue at all for this. isnt it suppose to be the same as the first one?

5)GOLD MEDAL- if 2 D letters come first and the 2 L letters come last, find how many different arrangement there are

D D _ _ _ _ _ L L
5P5 x 2P2 x 2P2. why i dont need to permute the letter D and the L?

6) 3 letters are selected at random from the word "BIOLOGY" . Find the number that the selection contains one letter O

2C1 x 5C2 = 20 , the answer given is 5C2 = 10 , why i dont have to write 2C1?

7)The sum of the first 5 terms of a G.P. is twice the sum of the terms from the 6th to the 15th inclusive. Show that r^5 = 1/2 (SQRT3 -1).

S5 = 2 (S15 - S5)
a(1-r^5)/ i-r = 2a/ 1-r [(1-r^15)-(1-r^5)]
1 - r^5 = 2(r^5 -r^15)
2r^15-3r^5+1=0
let u=r^5
2u^3 - 3u +1=0
i'm stuck here. what to do next?

[nona.m.nona]

1) ABCD is a parallelogram and the coordinates of A, B, C are (-8, -11), (1, 2) and (4, 3) respectively. Find the area of the parallelogram.

Use the Distance Formula to find the lengths of each base. Find the slope of the line containing either base; use this to find the perpendicular slope. Find the line equation for the perpendicular through one of the corners; find the point where this line intersects the other base (or the other base, extended). Find the distance between the two bases, and thus the height.

Plug the base lengths and the height into the area formula.

2). Points A and C have coordinates (-1, 2) and (9, 7) respectively. A rectangle ABCD has AC as a diagonal. Calculate the possible coordinates of B and D if the length of AB is 10 units.

from the question, i know that AB=CD=10. the equation of line AB = x^2 +y^2+ 2x -4y-97=0. the equation of line CD= a^2 +b^2-18 a- 14b +30=0. then what should i do next?

The line AB will be linear; what you have posted is a quadratic. How did you derive this? (One of the links in the previous response explains how to find linear equations from two points.)

As for finding the coordinates, try drawing the two points you have now, along with the line containing them. The rectangle is either to the one side of that line, or else to the other. So where must the other two points be?

3)If the five letters a, b, c, d, e are put into a hat, in how many ways could you draw two letters?

must we use permutation or combination? if permutation, why? why can't i use combination? 5C2?

Since nothing in the exercise says anything about ordering, combinations should be sufficient. However, you can see your book and your class notes; if your book, by default, defines "drawing" as "one at a time, with order taken into account", then you should use permutations.

4))3 balls are to be placed in 3 different boxes, not necessarily with 1 ball in each box. any box can hold up to 3 balls. find the number of ways the balls can placed if
i) they are all of the same colour
ii) they are all of different colours

i) what is wrong with my workings? 3C3 + (3C2 x 1C1) + 3C1?

You seem to be counting the ways to choose the balls, but not the ways in which you could then distribute them (in the chosen groupings) amongst the boxes.

ii) i have no clue at all for this. isnt it suppose to be the same as the first one?

If you have boxes A, B, and C and if you are placing three red balls in them, then you have the first situation. However, if you have boxes A, B, and C and if you are placing a red, a green, and a yellow ball in them, then you have the second situation. In the first case, "Box A has a red ball" is the same as "Box A has a red ball", since the balls are indistinguishable. In the second case, "Box A has a red ball" is different from "Box A has a green ball".

5)GOLD MEDAL- if 2 D letters come first and the 2 L letters come last, find how many different arrangement there are

D D _ _ _ _ _ L L
5P5 x 2P2 x 2P2. why i dont need to permute the letter D and the L?

How does "D" differ from "D"? How are they distinguishable?

6) 3 letters are selected at random from the word "BIOLOGY" . Find the number that the selection contains one letter O

2C1 x 5C2 = 20 , the answer given is 5C2 = 10 , why i dont have to write 2C1?

What was your logic for including "₂C₁"?

7)The sum of the first 5 terms of a G.P. is twice the sum of the terms from the 6th to the 15th inclusive. Show that r^5 = 1/2 (SQRT3 -1).

[i]S5 = 2 (S15 - S5)
a(1-r^5)/ i-r = 2a/ 1-r [(1-r^15)-(1-r^5)]

How did you arrive at this equation?

Using the formulation explained here:



Multiply by (1 - r)/a to get:







Then solve the polynomial for its rational root (which clearly does not apply, as a value for the common ratio, to this series). Solve the resulting quadratic for the other root, using the definition of "geometric progression" to pick the correct value.

[jigoku_snow]

sorry for the late reply.

The line AB will be linear; what you have posted is a quadratic. How did you derive this?

since i cant find the gradient so i use the distance formula. i let B=(x,y) and D=(a.b)

What was your logic for including "2C1"?

well, is because there are 2 'O' s. but i think i know why i dont have to write that, is because they are indistinguishable.

How did you arrive at this equation?

from a(1-r^5)/ 1-r = 2a [ (1-r^15/ 1-r) - ( 1-r^5 / 1-r )]
i simplify it .

Then solve the polynomial for its rational root

i think this question is way too far for me as i not yet learn polynomial.

[nona.m.nona]

The line AB will be linear; what you have posted is a quadratic. How did you derive this?

since i cant find the gradient so i use the distance formula. i let B=(x,y) and D=(a.b)

You have two points on a straight line. You can definitely find the slope ("gradient"). Please study the lesson in the link, provided earlier, to learn how.

Then solve the polynomial for its rational root

i think this question is way too far for me as i not yet learn polynomial.

You are working with polynomials, such as the quadratic you created for the first exercise and "1 - r¹⁵" here, so I am not sure what you mean when you say that you haven't yet learned about polynomials. Please study the lesson provided in the link to learn how to solve polynomials.