Given: Polygon STUVW is similar to polygon XYZAB, XY= 32, YZ= 36, ST= 4c+ 2, TU= 5c+ 2
Find : ST and TU
Definition of Similar Polygons: Two polygons containing vertices that can be paired so that the corresponding angles are congruent and the corresponding sides are in proportion.
ST and XY are corresponding sides.
TU and YZ are corresponding sides.
By the definition, corresponding sides are in proportion, therefore they can be expressed as a ratio: ST/XY = TU/YZ
Substitute the given values for XY, YZ, ST, TU into this equation:
ST/XY = TU/YZ
(4c+2)/32 =(5c+2)/36
Cross Multiply: 36(4c+2) = 32(5c+2)
Divide both sides by 4 (to simplify):
(36(4c+2))/4 = (32(5c+2))/4
9(4c+2) = 8(5c+2)
36c + 18 = 40c + 16
Subtract 36c from both sides
36c - 36c + 18 = 40c - 36c + 16
18 = 4c + 16
Subtract 16 from both sides:
18 - 16 = 4c + 16 - 16
2 = 4c
Divide both sides by 4: 2/4 = (4c)/4
Simplify: 1/2 = c or c=1/2
Now substitute c=1/2 into the equations for ST and TU:
ST= 4c + 2 = 4(1/2) + 2 = 2 + 2 = 4
ST=4
TU= 5c + 2 = 5(1/2) + 2 = 5/2 + 2 = 5/2 + 4/2 = 9/2
TU = 9/2
Check results: Substitute ST=4, TU=9/2 and XY= 32, YZ= 36 into the ratio:
ST/XY = TU/YZ
4/32 = (9/2)/36
Cross Multiply: 4(36) = 32(9/2)
144 = 16(9)
144 = 144 is correct, so values for XY and YZ are correct
To find the lengths of ST and TU, we can use the fact that the two polygons are similar. This means that the corresponding sides of the two polygons are proportional.
Let's set up the proportion:
ST / XY = TU / YZ
Substituting the given values, we have:
(4c + 2) / 32 = (5c + 2) / 36
Now, cross-multiply to eliminate the denominators:
36 * (4c + 2) = 32 * (5c + 2)
Simplifying the equation:
144c + 72 = 160c + 64
Rearranging and combining like terms:
144c - 160c = 64 - 72
-16c = -8
Divide both sides by -16:
c = -8 / -16
c = 1/2
Now we can find the values of ST and TU by substituting c = 1/2 into the given expressions:
ST = 4c + 2 = 4(1/2) + 2 = 2 + 2 = 4
TU = 5c + 2 = 5(1/2) + 2 = 5/2 + 2 = 5/2 + 4/2 = 9/2 = 4.5
Therefore, the values of ST and TU are: ST = 4 and TU = 4.5.