Question

# Prove the following statement.

The length of any one median of a triangle is less than half the perimeter of the triangle.

Hint:

- Triangle inequality theorem
- According to this theorem, in any triangle, sum of two sides is greater than third side,
- a < b + c
- b < a + c
- c < a + b

- The line segment which joins the vertex of triangle to midpoint of opposite side is called the ‘median’.
- Perimeter of triangle is sum of sides.
- Perimeter = a + b + c

- The line segment which joins the vertex of triangle to midpoint of opposite side is called the ‘median’.
- Perimeter of triangle is sum of sides.
- Perimeter = a + b + c

## The correct answer is: Hence, proved.

- The length of any one median of a triangle is less than half the perimeter of the triangle.

- Step 1:

Let,

Side AB = a

Side BC = b

Side AC = c

And median AM be M_{a}

- Step 2:

In △ABM,

AM is median so M is midpoint of BC

BM =

So, according to triangle inequality theorem,

M_{a} < a + ---- eq. 1

In △ACM,

AM is median so M is midpoint of BC

MC =

So, according to triangle inequality theorem,

M_{a} < c + ---- eq. 2

Add eq. 1 and eq. 2.

M_{a} + M_{a} < a + + c +

2M_{a} < a + b + c

- Final Answer:

Hence, proved.

_{a}

_{a}< a + ---- eq. 1

_{a}< c + ---- eq. 2

Add eq. 1 and eq. 2.

_{a}+ M

_{a}< a + + c +

_{a}< a + b + c

### Related Questions to study

### Select the three most common text features

### Select the three most common text features

### Solve 3.5x+19≥1.5x-7

### Solve 3.5x+19≥1.5x-7

### Determine whether each graph represents a function ?

An equation of the form y = f(x) is called a function if there is a unique value of y for every value of x. In other words, every value of x must have one value of y. When we draw a vertical, it cuts x axis at one point, we take that point to be x. If the vertical line cuts the graph at two points, then it gives two values of y for one value of x. This does not represent a function.

### Determine whether each graph represents a function ?

An equation of the form y = f(x) is called a function if there is a unique value of y for every value of x. In other words, every value of x must have one value of y. When we draw a vertical, it cuts x axis at one point, we take that point to be x. If the vertical line cuts the graph at two points, then it gives two values of y for one value of x. This does not represent a function.

### Choose the negative adjectives starting with ' u '

### Choose the negative adjectives starting with ' u '

### Describe the possible values of x.

### Describe the possible values of x.

### Write the solutions to the given equation.

Rewrite them as the linear-quadratic system of equations and graph them to solve.

### Write the solutions to the given equation.

Rewrite them as the linear-quadratic system of equations and graph them to solve.

### Describe the possible lengths of the third side of the triangle given the lengths of the other two sides.

5 inches, 12 inches

### Describe the possible lengths of the third side of the triangle given the lengths of the other two sides.

5 inches, 12 inches

### Identify the common noun in the given sentence

"The mice are afraid of a cat"

### Identify the common noun in the given sentence

"The mice are afraid of a cat"

### How many solutions does the given system of equations have?

y=x & y=x^{2}

### How many solutions does the given system of equations have?

y=x & y=x^{2}

### How can you determine whether the relationship between side length and area is a function?

If it is a function, say whether it is linear or non linear function ?

There are other ways to determine whether a function is linear or not, like, checking if the slope is equal between each of the points or if the equation can be written in the form of y = ax + b, where a and b are constants.

### How can you determine whether the relationship between side length and area is a function?

If it is a function, say whether it is linear or non linear function ?

There are other ways to determine whether a function is linear or not, like, checking if the slope is equal between each of the points or if the equation can be written in the form of y = ax + b, where a and b are constants.