Question

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

Answer 1

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

The parameters are as follows:

two rows of three dots spaced evenly

Two dots from one row will be chosen to form a triangle, while the third dot will be chosen from the opposite row.

As a result, the number of ways to select the dots is;

₃C₂ × ₃C₁ = 3 × 3 = 9 triangles

The same method can be repeated from the top row to produce 9 more triangles.

As a result, the total number of triangles is 18 triangles.

The number of different triangles discovered is as follows;

Given that triangles obtained from the top row are comparable to those obtained from the bottom row, we narrow the range of different triangles to 19 - 9 = 9 triangles.

The two neighboring dots of the three dots on the left and right of the lower row of dots make the same three triangles as the three dots on the top row of the nine triangles made by one dot on top and two dots on bottom.

As a result, because there are three sets of two dots generating nine triangles, each pair of dots can create three triangles, and as previously said, two pairs of dots from the three pairs produce the same triangles, resulting in the distinct triangle = nine i,e. = 9 - 3 = 6.

Answer 2

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

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.