Answer:
We are given two similar triangles, triangle ABC and triangle DEF, and some measurements of triangle ABC. We are also given that DF, one of the sides of triangle DEF, is equal to 15 units. Using this information, we are asked to find the difference between the lengths of sides AB and EF.
To solve the problem, we can first use the angle-angle similarity theorem to determine that the corresponding angles of the two triangles are equal. Therefore, angle DEF is equal to angle BCA, and angle ABE is equal to angle DFE.
Next, we can use the law of sines to find the length of side AB. The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is equal for all sides and angles in the triangle. Applying this to triangle ABC, we have:
AB/sin(73) = BC/sin(48)
Substituting the value of BC as 19 units, we can solve for AB to get:
AB = 22.78 units
Similarly, we can use the law of sines to find the length of side EF. Since angle DFE is equal to angle ABE, we can use the same ratio as above to get:
EF/sin(73) = DF/sin(48)
Substituting the value of DF as 15 units, we can solve for EF to get:
EF = 18.20 units
Finally, we can subtract EF from AB to get:
AB - EF = 22.78 - 18.20 = 4.58 units
Therefore, the difference between the lengths of sides AB and EF is 4.58 units.