How many distinct triangles can be drawn using three of the dots below as vertices?

The number of distinct triangles that can be drawn using the dots = 6

Step-by-step explanation:

The parameters given are;

Two rows of three evenly spaced dots

To form a triangle, two dots will be selected from 1 row while the third dot will be selected from the other row

The number of ways of selecting the dots are therefore;

₃C₂ × ₃C₁ = 3 × 3 = 9 triangles

The same procedure can be done from the top row to give another 9 triangles

Which gives the total number of triangles = 18 triangles

The number of distinct triangles are found as follows;

Given that triangles obtained from the top row are similar to those of the bottom row, we reduce the range from which the distinct triangles can be found to 19 - 9 = 9 triangles

Of the 9 triangles formed by one dot on top and two dots on the bottom, the two adjacent dots of the three dots which are on the left and on the right of the lower row of dots, form the same three triangles with the three dots on the top row

Therefore, since there are 3 sets of two dots forming 9 triangles, each pair of dots can form 3 triangles, and as mentioned, 2 pairs of dots of the 3 pairs form the same triangles making the distinct triangle = 9 - 3 = 6.

your exact question is kind of difficult to understand, so maybe an example will explain what you're supposed to do? i'll work through the first problem on the worksheet.

suppose you have a tray of 652 brownies, and you have 5 friends that you want to feed without picking favorites. how many brownies can you dole out to each of your friends in equal amounts? how many will be left over for you to eat?

what you're doing here is finding a way to arrange the number 652 as the sum of integer multiple of 5 and some remainder:

here ("q" for quotient) is the number of brownies each of your friends get, and is the remainder. that's the basic idea behind division.

we break up the number 652 into parts so that the leading digit stands on its own:

652 = 600 + 52

first we check if it's possible to hand out exactly 100 brownies to each friend. with 5 friends, this means we want to know if we can take out 500 brownies from the tray. we can do this 1 time because we have 600 brownies to take from, and when we do that we're left with 100 + 52 brownies to divvy up. so we know that

652 = 1*500 + 152

next we want to know if it's possible to hand out exactly 10 brownies per friend. if we can, we have to remove 50 brownies from the tray each time until we can't. we have 152 brownies left, and we know that 3(50) = 150, so

152 = 3*50 + 2

now we have 2 brownies left in the tray. is it possible to hand 1 out to each of your 5 friends? no, you'd need a minimum of 5 brownies to do that. so the last 2 are yours.

putting everything together, we found that

652 = 1*500 + 3*50 + 2

and we can rearrange this to write

652 = 5*100 + 5*30 + 2

652 = 5(100 + 30) + 2

652 = 5*130 + 2

so and .

you might be more accustomed to the table depicting the algorithm. that's just a compact way of organizing what we've done above. you can consult the image i've attached below.