Mathematics

Key unit competence

By the end of this unit, I will be able to use Thales’ theorem to solve the problem related to similar shapes and determine their lengths and area.

Unit outline

• Midpoint theorem

• Thales’ theorem and its converse

• Application of  Thales’ theorem in calculating lengths of proportions segments (In triangles and Trapeziums)

5.1 Midpoint theorem

Activity 5.1

1. Using a ruler, draw a line segment AB of length 10 cm.

2. Mark Point C 5 cm from A towards B at the midpoint of the line AB. Measure and compare the lengths AC and CB. What can you say about these two line segments AC and CB?

3. What Fraction does segment AC represent in terms of length AB?

Midpoint is defined as the point halfway between the endpoints of a line segment. A midpoint divides a line segment into two equal segments. Consider Fig. 5.1.

In figure 5.1 above, two people are 10 metres apart. They want to meet at the midpoint. The midpoint is half way from each person. The concept of midpoint of lines can be extended to triangles and trapezia to establish the proportions between parallel line segments.

This is summed up in what is known as midpoint theorem.

1. With the aid of a ruler, construct the above triangle accurately.

2. Measure and mark points D and E, the midpoints of AC and BC respectively. 3. Join D to E with a straight line as shown in Figure 5.3. Measure the length of the line DE 4. Compare the lengths of DE and AB and state the relationship. 5. Draw dotted perpendicular lines from points D and E to intersect line AB as shown in Figure 5.3. Measure and compare their lengths. What do you notice?

6. From your results in step 5, are the line segments DE and AB parallel or not

In Fig 5.4, Point S is the mid-point of the line segment PQ and T is the mid-point of the line segment PR. Hence the line segment PS is equal to SQ, and the line segment PT is equal to TR. 

When the two mid-points S and T are joined together, they form the line segment ST. From the results of Activity 5.2, ST is parallel to QR. By measuring, we see that the length of ST is half the length of QR

These facts are summarised in midpoint theorem that states:

1. The straight line through the midpoints of two sides of the triangle is parallel to the third side of the triangle.

2. The length of the segment joining the midpoints of the sides of the triangle is half the length of the third side which are parallel to it.

The midpoint theorem is also extended to trapezia.

Activity 5.3

In trapezium ABCD below,  

AB = 4 cm,

BC = 6 cm and  

CD = 5 cm and

AD = 8 cm.

1. With the aid of a ruler, construct the above trapezium accurately

2. Measure and mark points E and F, the midpoints of AB and DC respectively. Join E to  F with a straight line as shown in Fig. 5.5.

3. Draw dotted perpendicular lines from points E and F to intersect line AD. Measure and compare their lengths. What do you notice?
 

4. From your results in step 3, are the line segments DE and AB parallel or not?

In the trapezium WXYZ, WX is parallel to ZY. ZY is the base of the trapezium. Point A and B are the midpoints of ZW and YX respectively.

Therefore, AB is parallel to ZY.

These facts are summarised in the midpoint theorem for trapezia that states that: The line through the midpoint of two non-parallel sides of a trapezium is parallel to the base of the trapezium. Note: Applying the midpoint theorem in triangle ZWY, you should note that

AC=1/2 ZY, C is the midpoint of WY hence WC=CY.

Solution By midpoint theorem, DE = 1/2 BC
We get 4x =
1/2(6x + 4)
8x = 6x + 4

8x – 6x = 4

2x = 4     

so x = 2

Example 5.2

In the Trapezium below, E is the midpoint of AD, find the value of x, given that

AB = 6x2 – 36 and EF = 2x2 – 3x.
 

Exercise 5.1

1. In the figure below, find the value of x given that PQ is parallel to AC.

                             

2. In the figure below, find the value(s) of x given that PQ is parallel to AB.

                             

3. In the trapezium below,    FG = 2x2 – 5x and AB = 14 cm. FG is parallel to AB.

                                     

Find the values of x that can balance the conditions of the parallel sides.

4. In the Fig. 5.13, PQ is parallel to BA, AC is 6 cm. Triangle BCA is rightangled at C.

                                        

                                               Find the value of x.

5. Find the values of x in the diagrams below.

                                           

6.  Find the value of y in the triangle PQR if point M is the midpoint of the side PQ and N is the midpoint of the side PR given that QR=30 cm.

                                           

5.2  Thales’  theorem

5.2.1 Thale's theorem in triangles

Activity 5.4

1.

a)  Write the following as    proportions.

b) Find the values of letters   given in the ratios,

  i) 4:6 = c:3 

ii) 5:4  =15:x

2. a) Draw triangle ABC with   dimensions of your choice.

   b) Draw a line DE parallel to   side AB.

                           

 c)  Measure and record the  lengths of the line segments AD, DC, BE and EC.

d)  Find and compare the ratios AD/DC and BE/CE. What do you notice?

Remember, a ratio is a way of comparing two or more quantities of the same kind. Given that a:b = c:d, we can write this as a proportion of two equal ratios as ab=cd. In step 2 of activity 5.4, you should have noticed that AD/DC = BE/CE.

This fact is summarised in Thales’ theorem which states that: If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides the two sides in the same ratio. For example, in triangle PQR below ST is parallel to PQ.

                               

5.2.2 Thales’ theorem in Trapezia

Activity 5.5

1. Draw three parallel lines AB, CD and EF.

2. Draw two lines that pass through the parallel lines (Transverse lines) as shown in Fig. 5.19

        

What is the name of figure EABF?

3. Measure and record the lengths of the line segments AC, CE, BD and DF. 4. Find and compare the ratios AC/CE  and BD/DF . What do you notice?

In activity 5.5, we drew three parallel lines and two transversals through them (figure 5.19). Figure EABF is a trapezium.

5.3 The converse of  Thales’ theorem

Activity 5.6

1.  Draw triangle ABC with dimensions of your choice.

2.  Draw a line XY through side AC and BC; ensure the line is not parallel to AB.

                                    

3.  Measure and record the lengths of the line segments AX, XC, BY and YC.

4.  Find and compare the ratios AX/XC  and BY/YC . What do you notice

In activity 5.6, you should have noticed that the ratio AX/XC  ≠ BY/YC

This observation is the converse of  Thales’ theorem, which can be summed up as follows: “If a line intersects two sides of a triangle and is not parallel to the third side, then it does not divide the sides in the same ratio.

5. Copy and complete the proportions in figure 5.29.

                      

 (b) Calculate the values of x and y.

6. Find the values of x in the Fig. 5.30.

                          

7. In the Fig. 5.31 below, find the value of x in:

                           

8. Find the values of x in the figures below.

              

Unit Summary

1. Midpoint: It is the point halfway between the endpoints of a line segment. For example, if AX = XB then X is the midpoint of line AB.

2. Thales' theorem states that if a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides the two sides in the same ratio. For instance, in triangle ABC in Fig. 5.34.

                  

Thales' theorem states that; BX/XA = CY/YA .

3. The converse of Thales theorem states that if a line intersects two sides of a triangle and is not parallel to the third side, then it does not divide the sides in the same ratio.

4. The midpoint theorem for trapezia it states that the line through the midpoint of two non-parallel sides of a trapezium is parallel to the base of the trapezium.

5. The midpoint theorem states that:

a) The straight line through the midpoints of two sides of the triangle is parallel to the third side of the triangle.

b)  The length of the segment joining the midpoints of the sides of the triangle is half the length of the third side which are parallel to it.

6. A ratio is a way of comparing two or more quantities of the same kind. For example 1:2, 3:4 are ratios.

Unit 5 test 1.

In the figure 5.35, ACB is a rightangled triangle. The bisector of AC and BC is parallel to BA.
                           

Find the value of x.

3. Given that P and Q are the midpoints of lines AB and BC respectively in the Fig. 5.37.

                       

4. Fig. 5.38 is a trapezium. AM is half of AB and CN is half of CD. Given that AB = 8 cm and CD=10 cm.
                      

a) Determine the values of n, x and y.

b) Express the following as ratios to their simplest forms.

i) MB:CD  ii)  AM:DN

5. The right-angled triangle ABC in  Fig. 5.38 has E as the midpoint of line segment AB and F as the midpoint of line segment of BC. Given that  AB=9 cm and EF=6 cm.

                                 

 Find: 

a) The value of y

b) Length CF 

c) Area of triangle EBF and  ABC

6. In the Figure 5.40, AB = 2.4 cm,  AC = 3.6 cm, BC = 3 cm and  BB′ = 3.6 cm