## Key Concepts

- Midsegment Theorem.
- Midsegment Theorem to find lengths.
- Placing figure in coordinate plane

### Introduction

In the previous session, we learned about the properties of triangles and in the previous classes we learned about finding distances, simplifying expressions, solving equations and inequalities.

Now we will learn about the midsegment theorem and coordinate proof.

Have you heard about the midsegment theorem?

Do you have any idea about segments?

### Midsegment Theorem

A triangle has three sides.

A segment which connects the midpoints of two sides of a triangle are called midsegments of a triangle.

The midsegment theorem states that the midsegment of two sides of a triangle is parallel to the third side and the length of the midsegment is half the length of the third side.

### Drawing a Midsegment

How do you draw a midsegment?

Let us see this.

Step 1:

At first draw a triangle by maintaining all the properties of a triangle.

Step 2:

Then find the midpoints for any two sides of the triangle.

Step 3:

Now connect these midpoints using a line segment.

This segment is called the midsegment.

### Finding Lengths

Now we use the midsegment theorem to find the lengths.

Let us see an example.

Example:

Given a triangle BCD, where GF, GE, ED are the midsegments.

Find the value of GF, if BD = 24 inches.

If the

∠GFE=35, then ∠GFE=35°, then find ∠FED.

Explanation:

From the midsegment theorem,

GF is the midsegment of BD, so GF = half the length of BD = ½ * 24 = 12 inches

∠GFE=35

GF ‖ BD, similarly BC ‖ EF

So, n∠FED = 35°

#### Placing a Figure in a Coordinate Plane

How can we place a figure in a coordinate plane?

**Explanation: **

We can place a figure in a coordinate plane by considering the first quadrant of the coordinate plane.

It is easy to find the length if we consider the origin as one vertex and

we place one vertex on the x-axis and one more on the y-axis.

Let us see some figures on the coordinate plane.

**Example 1: **

How to draw a rectangle on a coordinate plane?

**Explanation: **

We consider the origin as one vertex.

Length is 12 units, which is plotted on the x-axis.

Breadth is 9 units, which is plotted on the y-axis.

The other vertex will be (12, 9)

Now we draw the lines to connect all vertices, thus the figure obtained is a rectangle.

**Example 2: **

How to draw a triangle on a coordinate plane?

**Explanation: **

We consider origin as one vertex.

We consider 15 on the y-axis,

The other vertex as (10, 8).

Now we draw the lines to connect all vertices.

Thus, the figure obtained is a triangle.

#### Constructing Midsegment on a Coordinate Plane

First, we construct a triangle on the coordinate plane.

(See the graph)

The vertices are (0, 0) (0, 15) (10, 8)

Then we draw the midsegments for this triangle by finding the midpoints.

Mid points will be

(0+102,15+82) , (0+102,0+82)

= (5, 11.5) (5, 4)

Now we connect these two midpoints to form a midsegment.

This midsegment is parallel to the base of the triangle.

### Application of Variable Coordinates

If an isosceles right triangle of base length k units is placed in a coordinate plane,

find the hypotenuse and midpoint of hypotenuse.

Explanation:

We consider the origin as one vertex.

We consider k units on the y-axis,

We consider k units on the x-axis.

Then we construct the triangle,

Now the length of the hypotenuse is the distance from (0, K)

(K, 0).

Hypotenuse = √(0−K)^{2}+(K−0)^{2}

= √K^{2}+K^{2}

=√K^{2}

By using the midpoint formula, we find the midpoint of the hypotenuse

Midpoint = (0 + K/2, K + 0/2)

(K/2, K/2)

### Midsegment Theorem and Its Proof

In the previous slides, we stated midsegment theorem, now we will check its coordinate proof

Midsegment theorem:

The midsegment of two sides of a triangle is parallel to the third side and the length of the midsegment is half the length of the third side.

Proof:

Given AB is the midsegment of ΔXYZ,

we need to prove that AB ‖ YZ and AB = ½ YZ

we see the graph for ΔXYZ

AB ‖ YZ:

The x-coordinates of A and B are the same, so the slope is the same.

Similarly, the x-coordinates of Y and Z are on the y-axis where x = 0, so the slopes of both AB and YZ are the same.

As slopes are the same AB ‖ YZ

AB = ½ YZ:

X (2x, 2z) Y (0, 2y), Z (0, 0) are the vertices of the triangle XYZ.

A is the midpoint of XY

We find A by using the mid-point formula.

A = (0+2x/2 ,2y+2z/2)

=(x, y+z)

Therefore A =(x, y+z)

B is the midpoint of XZ

We find B by using the midpoint formula.

B = (0+2x/2, 0+2z/2)

=(x, z)

Therefore B =(x, z)

Now,

YZ = √(0−0)^{2} + (0−2y)^{2}

* *(By distance formula)

= √(2y)^{2}

=2y

AB = √(x−x)^{2} + (y+z−z)^{2}

* *(By distance formula)

= √(y)^{2}

=y

AB = ½ YZ:

AB = yAB = y

= 1/2(2y)

=1/2(YZ)

Therefore AB = ½ YZ.

Hence proved.

### Midsegment Theorem Extension

We can extend the midsegment theorem by proving the midsegment theorem for all sides of the triangle.

We can prove both the properties parallel and half of the length property for all the three sides of the triangle if we proved for one side.

**Real Life Example (Disco Ball)**

Given a disco ball, if in its AD ≅ DB, AE ≅ EC, prove that DE ‖ BC.

Explanation:

Given a disco ball, if in its AD ≅ DB, AE ≅ EC

We need to prove that DE ‖ BC.

From the given information, D is the midpoint of AB and E is the midpoint of AC.

Therefore, DE forms a midsegment.

So, DE ‖ BC from the midsegment theorem.

## Exercise

- If DE = 5 inches is the midsegment of a triangle ABC, find the length of BC.

- Place the below figures in a coordinate plane in a convenient way.
- A rectangle of length of x units and breadth of y units.
- Isosceles right triangle of length 9 units, find the hypotenuse.
- Place a triangle in two different ways in a coordinate plane, find the hypotenuse in both the cases. Check whether the value of hypotenuse is the same or not in both cases.
- Find the length and slope of the triangle with vertices (a, 0), (0, 0) (a, b). In a triangle ABC- D, E, F are the midpoints.

- If AB = 3x + 15, DE = 5x + 4, what is the length of AB?
- If DF = 9y + 4, BC = 3y + 13, what is the length of DF?
- If the midpoints of each side of a triangle are (4, 0) (5, 8) (9, 12), find the perimeter of the triangle formed by the midsegments.
- If the midpoints of each side of a triangle are (5, 2) (12, 4) (9, 9), find the perimeter of the triangle formed by the midsegments.
- If the midpoints of each side of a triangle are (5, 2) (12, 4) (9, 9) and forms a triangle. Find the original triangle.
- In a classroom, a projector is placed to explain midsegments, a student of height of 4 feet is standing in between a projector and the wall on which projection is made. The edge of the projector light reaches the top of his head. What is the height of his shadow?

### Concept Map

### What have we learned

- Midsegment Theorem.
- Mid segment Theorem to find lengths.
- Placing figure in coordinate plane.

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