Perimeter = 7 + 24 + 25 = 56cm
I had to make ten more right angled triangles that follow the rule a2+b2=c2
The following triangles are what I made and I also found out the area and the perimeter of them.
9² = 81
40² = 1600
41² = 1681
81+ 1600 = 1681 C=41
Area = 9 x 40 = 360 2 = 180 cm² A=9
Perimeter = 9 + 40 + 41 = 90cm
11² = 121
60² = 3600
61² = 3721 C=61
121 + 3600 = 3721 A=11
Area = 11 x 60 =660 2 = 330cm B=60
Perimeter =11+ 60 + 61 = 132 cm
13² = 169 C=85
84² = 7056 A=13
85² = 7725
169 + 7056 = 7725
B=84
Area =13 x 84 = 1092 2 = 546 cm²
Perimeter = 13 + 84 + 85 = 182cm
15² = 225 C=113
112² = 12544 A=15
113² = 12769
225 + 12544 = 12769
B=112
Area = 15 x 112 = 1680 2 = 840 cm²
Perimeter = 15 + 112 + 113 = 240cm
17² = 289
144² = 20736 C=145
145² = 21025 A=17
289 + 20736 = 21025
B=144
Area = 17 x 144 =2448 2 = 1224 cm²
Perimeter =17 + 144 + 145 = 306cm
19² = 361
180² = 32400 C=181
181² = 32761 A=19
361+ 32400 = 32761
B=180
Area = 19 x 180 = 3420 2 = 1710 cm²
Perimeter =19 + 180 + 181 = 380 cm
C=221
21² = 441
220² = 48400 A=21
221² = 48841
441 + 48400 = 48841 B=220
Area = 21 x 220 = 4620 2 =2310 cm²
Perimeter =21+ 220 + 221 = 462 cm
23² = 529 C=265
264² = 69696 A=23
265² =70225
529 + 69696 = 70225
B=264
Area = 23 x 264 = 6072 2 = 3036 cm²
Perimeter = 23 + 264 + 265 = 552cm
25² = 625 C=313
312² = 97344 A=25
313² = 97969
625 + 97344 = 97969
B=312
Area = 25 x 312 = 7800 2 = 3900cm²
Perimeter = 25 + 312 + 313 = 682cm
27² = 729 C=365
364² = 132496 A=27
365² = 133225
729 + 132496 = 133225
B=364
Area = 27 x 364 = 9828 2 = 4914cm²
Perimeter = 27 + 364 + 365= 756cm
Rules: 2n+1 2n²+2n 2n²+2n+1 4n²+6n+2 2n³+3n²+n
I then found out the rule each of the columns follow. How I worked the sequence out it shown below.
This is how I found out the rule the first column follows (opposite line):-
Rule = 2n +1
This is how I found out the rule the second column follows (adjacent line):-
A
B
Rule = 2n²
A-B
Rule = 2n
So the rule that the adjacent line column follows is = 2n²+2n
This is how I found out the rule the third column follows (hypotenuse line):-
A
B
Rule = 2n²
A-B
Rule = 2n+1
So the rule that the hypotenuse line column follows is = 2n²+2n+1
This is how I found out the rule the Fourth column follows (perimeter):-
A
B
Rule = 4n²
A-B
Rule = 6n+2
So the rule that the perimeter column follows is = 4n²+6n+2
This is how I found out the rule the Fifth column follows (area):-
A
B
Rule = 2n³
A-B
C
(A-B) –C
Rule = n
So the rule that the area column follows is = 2n³+3n²+n
I then tried to change the results to even numbers and checked if it still followed the rule a²+b² =c².
I got the first equation which was 3, 4, 5 and I times each number by 2. I times each number by to because then it will make the first number even and I checked if it worked. Here is my working out to see if I changed them into even would it still follow the rule:-
3 x 2 =6 4 x 2 =8 5 x 2 =10
My new equation was 6, 8, 10
6²=36
8²=64
10²=100 C=10
36+64 =100 A=6
Area = 6 x 8 =48 2 =24 cm² B=8
Perimeter =6+8+10 = 24cm
This equation does work.
I then times the next equation by 2 which gave me 10,24,26
I checked if this equation worked like the other one and I don’t the same to the other four.
10² = 100
24² = 576 C=26
26² = 676 A=10
100 + 576 = 676
Area = 10 x 24 =240 2 = 120cm² B=24
Perimeter = 10 + 24 + 26 =60cm
14² = 196 C=50
48² = 2304 A=14
50² = 2500
196 + 2304 = 2500
B=48
Area =14 x 48=672 2 =336 cm²
Perimeter = 14 + 48 + 50 =112cm
18² = 324
80² = 6400 C=82
82² = 6724 A=18
324 + 6400 = 6724
B=80
Area = 18 x 80 =1440 2 =720 cm²
Perimeter = 18 + 80 + 82 =180cm
22² = 484 C=122
120² = 14400 A=22
122² = 14884
484 + 14400 = 14884
B=120
Area = 22 x 120 = 26402 =1320 cm²
Perimeter = 22 + 120 + 122 = 264cm
Rule: 4n+2 4n²+4n 4n²+4n+2 8n²+12n+4 8n³+12n²+4n
I then found out the rule each of the column in the table above which is similar to the first table but I found out a way to make all the equations even. All u had to do is time each of the numbers in each equation by 2 to get them to even and I checked if it follows the rule a²+ b² =c².and I found out it does. I then had to find out the rule that each of the columns follow. My working out is shown below.
This is how I found out the rule the first column (opposite line):-
Rule = 4n+2
This is how I found out the rule the second column follows (adjacent line):-
A
B
Rule = 4n²
A-B
Rule = 4n
So the rule that the adjacent line column follows is = 4n²+4n
This is how I found out the rule the third column follows (hypotenuse line) :-
A
B
Rule = 4n²
A-B
Rule = 4n+2
So the rule that the hypotenuse line column follows is = 4n²+4n+2
This is how I found out the rule the Fourth column follows (perimeter) :-
A
B
Rule = 8n²
A-B
Rule = 12n+4
So the rule that the perimeter column follows is = 8n²+12n+4
This is how I found out the rule the Fifth column follows (area) :-
A
B
Rule = 8n³
A-B
C
(A-B) –C
Rule = 4n
So the rule that the area column follows is = 8n³+12n²+4n
I then times the first results by 3 and also checked if it follows the rule a² + b² = c².
My working out is shown below:
9² = 81
12² = 144
15² = 225
81 + 144 = 225 C=15
A=9
Area = 9 x 12 = 108 2 =54cm²
Perimeter =9 + 12 + 15 = 36cm
B=12
15² = 225
36² = 1296 A=15 C=39
39² = 1521
225 + 1296 = 1521
B=36
Area = 15 x 36 = 540 2 = 270 cm²
Perimeter =15+36+39 = 90cm
21² = 441
72² = 5184 A=21 C=75
75² = 5625
441 + 5184 = 5625
B=72
Area = 21 x 75 = 15122 =756 cm²
Perimeter =21+72+75 = 168cm
27² = 729
120² = 14400 A=27 C=123
123² = 15129
729 + 14400 = 15129
B=120
Area = 27 x 120 = 32402 =1620 cm²
Perimeter = 27 + 120 + 123 = 270cm
33² = 1089
180² = 32400 C=183
183² = 33489 A=33
1089 + 32400 = 33489
B=180
Area = 33 x 180 = 59402 = 2970 cm²
Perimeter = 33 + 180 + 183 = 396cm
I then plotted these result on a table:
I then found the rule each of the columns follow. My working out is shown below:
This is how I found out the rule the first column (opposite line):-
Rule = 6n+3
This is how I found out the rule the second column follows (adjacent line):-
A
B
Rule = 6n²
A-B
Rule = 6n²+6n
So the rule that the adjacent line column follows is = 6n²+6n
This is how I found out the rule the third column follows (hypotenuse line) :-
A
B
Rule = 6n²
A-B
Rule = 6n²+6n+3
So the rule that the hypotenuse line column follows is = 6n²+6n+3
This is how I found out the rule the Fourth column follows (perimeter) :-
A
B
Rule = 12n²
A-B
Rule = 12n²+18+6
So the rule that the perimeter column follows is = 12n²+18+6
This is how I found out the rule the Fifth column follows (area) :-
A
B
Rule = 18n³
A-B
C
Rule = 27n²
(A-B) –C
Rule = 18n³+27n²+9n
So the rule that the area column follows is = 18n³+27n²+9n
I then times the first results by 4 and checked if it follows the rule a² + b² = c². I done this so I can compare the four different scale factors and see if I can find out the x term.
My working out is shown below:
12² =144
16² =256
20² = 400 A=12 C=20
144 + 256 = 400
Area = 12 x 16 = 192 2 = 96cm²
Perimeter =12 + 16 + 20 = 48cm B=16
20² = 400
48² = 2304
52² = 2704 A=20 C= 52
400 + 2304 = 2704
Area = 20 x 48 = 960 2 = 480cm²
Perimeter =20 + 48 + 52 = 120cm B=48
28² = 784
96² = 9216
100² = 10000
784 + 9216 = 10000 A= 28 C= 100
Area = 28 x 96 = 2688 2 = 1344cm²
Perimeter = 28 + 96 + 100 = 224cm B=96
36² = 1296
160² = 25600
164² = 26896 A=36 C=164
1296 + 25600 = 26596
Area = 36 x 160 = 5760 2 = 2880cm²
Perimeter = 36 + 160 + 164 = 360cm B=160
44² = 1936
240² = 57600
244² = 59536 A= 44 C=244
1936 + 57600 = 59536
Area = 44 x 240 = 10560 2 = 5280cm²
Perimeter = 44 + 240 + 244 = 528cm B=240
I then plotted these result on a table:
I will now try to find out the rule for each of the sides like I did for when I had the scale factor as 2 and 3:
This is how I found out the rule the first column (opposite line):-
Rule = 8n + 4
This is how I found out the rule the second column follows (adjacent line):-
A
B
Rule = 8n²
A-B
Rule = 8n²+8n
So the rule that the adjacent line column follows is = 8n²+8n
This is how I found out the rule the third column follows (hypotenuse line) :-
A
B
Rule = 8n²
A-B
Rule = 8n² + 8n + 4
So the rule that the hypotenuse line column follows is = 8n² + 8n + 4
This is how I found out the rule the Fourth column follows (perimeter) :-
A
B
Rule = 6n²
A-B
Rule = 6n² + 24n + 8
So the rule that the perimeter column follows is = 6n² + 24n + 8
This is how I found out the rule the Fifth column follows (area) :-
A
B
Rule = 32n³
A-B
C
Rule = 48n²
(A-B) - C
By Zakir Md. Hussain