Q19. - Geometry

Question

State how many segments can be drawn between the points in each figure. No three points are collinear.

a. 3 points $\underset{¯}{?}$ segments.

b. 4 points $\underset{¯}{?}$ segments.

c. 5 points $\underset{¯}{?}$ segments.

d. 6 points $\underset{¯}{?}$ segments.

e. Without making a drawing, predict how many segments can be drawn between seven points, no three of which are collinear.

f. How many segments can be drawn between $n$ points, no three of which are collinear?

Step-by-Step Solution

Answer

1. The number of segments that can be formed between the 3 points is 3.
2. The number of segments that can be formed between the 4 points is 6.
3. The number of segments that can be formed between the 5 points is 10.
4. The number of segments that can be formed between the 6 points is 15.
5. The number of segments that can be formed between the 7 points is 21.
6. The number of segments that can be formed between the $n$ points is $\frac{n\left(n-1\right)}{2}$

Part a. Step 1. Definition of segment.

A segment is a portion of a line which is bounded by the two end points.

Part a. Step 2. Observe the given diagram.

The given diagram is:

Part a. Step 3. Count the number of segments formed between the 3 points.

The number of segments that are formed between the 3 points are 3.

Part b. Step 1. Definition of segment.

A segment is a portion of a line which is bounded by the two end points.

Part b. Step 2. Observe the given diagram.

The given diagram is:

Part b. Step 3. Count the number of segments formed between the 4 points.

The number of segments that are formed between the 4 points are 6.

Part c. Step 1. Definition of segment.

A segment is a portion of a line which is bounded by the two end points.

Part c. Step 2. Observe the given diagram.

The given diagram is:

Part c. Step 3. Count the number of segments formed between the 5 points.

The number of segments that are formed between the 5 points are 10.

Part d. Step 1. Definition of segment.

A segment is a portion of a line which is bounded by the two end points.

Part d. Step 2. Observe the given diagram.

The given diagram is:

Part d. Step 3. Count the number of segments formed between the 6 points.

The number of segments that are formed between the 6 points are 15.

Part e. Step 1. Formula to calculate the number of segments that are formed between m points no three of which are collinear.

The formula to calculate the number of segments that can be formed between $m$ points no three of which are collinear is:

$No.ofsegments=\frac{m\left(m-1\right)}{2}$.

Part e. Step 2. Substitute 7 for m in the above formula to calculate the number of segments that can be formed between 7 points.

Therefore,

$No.ofsegments=\frac{7\left(7-1\right)}{2}$

Part e. Step 3. Solve the above expression to calculate the number of segments that can be formed between 7 points.

$$\begin{gathered} No.ofsegments=\frac{7\left(6\right)}{2} \\ =\frac{42}{2} \\ =21 \end{gathered}$$

Therefore, the number of segments that can be formed between the 7 points are 21.

Part f. Step 1. Formula to calculate the number of segments that are formed between m points no three of which are collinear.

The formula to calculate the number of segments that can be formed between $m$ points no three of which are collinear is:

$No.ofsegments=\frac{m\left(m-1\right)}{2}$.

Part f. Step 2. Substitute n for m in the above formula to calculate the number of segments that can be formed between n points.

Therefore, 

$No.ofsegments=\frac{n\left(n-1\right)}{2}$

Part f. Step 3. write the number of segments that can formed between n points.

Therefore, the number of segments that can be formed between the $n$ points are $\frac{n\left(n-1\right)}{2}$.