Lesson 8: Triangles with the Same Area

 

 

 

 

 

 

 

 

 


Is the area of triangle ABC the same as the area of triangle ACD, if BC = CD, BD = 8m and h = 5m? Use a red dotted line to show the height of the triangle.

A
 

Since the height and the base of triangle ABC are the same as the height and the base of the triangle ACD, the area of the two triangles is the same.

 

    Since BD = 8m, BC = CD = 4m.

 
 
 

 

 

 

 

 

 

 


      S ΔABC = ½ (BC x h) = ½ (4 x 5) = ½ x 20 = 10 m 2 ,    

      S ΔACD = ½ (CD x h) = ½ (4 x 5) = ½ x 20 = 10 m 2 .

 

Exercise 8: Triangles with the same area: answer the following.

(1)  Is the area of triangle ABC the

     same as the area of triangle

     ACD, if BC = CD, BD = 4 m

     and h = 3?

 
 
 

 

 


 

                     A

 

 

 

B       C         D           E
 
 


Isosceles Triangle: h 

S ΔABC = ________  m 2     

 

S ΔACD = _________ m 2

 

S Δ ADE = _________ m 2

 

 
 
 

 

 

 

 

 

 


 

 

 

Lesson 9: Critical Thinking: Dividing a Triangle into Smaller Triangles of the Same Area

    A

 

 


                      B                                             C

                                    A

 

 

 

B          D       E         F        C
 
 


(1) Divide the base into 4 equal parts at

     points D, E, and F. Draw lines from

     A to D, E, and F. There are four

     triangles now.

         ΔABD, ΔADE, ΔAEF, and ΔAFC

 

     Since the height of the four triangles is the same, and BD = DE = EF = FC,

 

                            S  ΔABD = S ΔADE = S Δ AEF = S ΔAFC

 

                                 A

                 E                  

                                        F

 

B                   D                   C
 
(2) Divide AB in half at point E.

     Divide BC in half at point D.

     Divide AC in half at point F.

     Connect DE, DF, and AD.

     There are four triangles now.

     ΔAED, ΔADF, ΔBED, and   ΔDFC

 

     Since the height of the triangles ADC and ADB

is the same, and BD = DC,

 

           S ΔADC = S ΔADB

 

     Since the height of the triangles ADE and DBE

is the same, and BE = EA,

1

2
 

1

2
 
 


           S ΔADE =  S ΔDBE =        S ΔADC  =         S ΔADB

 

     Since the height of the triangles ADF and DCF

is the same, and CF = FA,

1

2
 

1

2
 
 


            S  ΔADF =  S ΔDCF =         S ΔADB =        S  ΔADC  

 

     The areas of the four triangles are the same.

Exercise 9: Critical thinking: divide triangles into smaller triangles of the same area.

 

(1)   Is there more than one method of dividing the following triangle into four triangles with the same area?

 

                                  A

 

 

     B                                           C

 

 

(2)   Can you divide the following triangle into three triangles, where the ratio of their areas is 1 : 2 : 3?

 

                                    A

 

 


    B                                            C

 

 

(3)   Can you divide the following triangle into three triangles, where the ratio of their areas is 1 : 3 : 5?

 

                                    A

 

 


    B                                            C

 

 

 

(4) If BE = 3 AE, BC = 2BD, and  S ΔBDE = 1 m 2, then S ΔABC =?

                              A

                        E

 

              

 

    B                                 C

                       D