Answer: Therefore, the length DE is 12.5.
Explanation:
To find the length DE in triangle ACB, we can use the angle-angle similarity theorem. According to the theorem, if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
In this case, angle ABC is congruent to angle DEC. Therefore, triangles ABC and DEC are similar.
Given that CD = 35 and CA = 84, we can set up a proportion between the corresponding sides of the triangles:
AB / AC = DE / DC
Substituting the known values:
AB / 84 = DE / 35
To find the length DE, we can solve this proportion. Cross-multiplying:
AB * 35 = 84 * DE
Simplifying:
35AB = 84DE
Dividing both sides by 35:
AB = (84 / 35) * DE
Simplifying further:
AB = 12/5 * DE
We are also given that AB has a length of 30. Substituting this value:
30 = 12/5 * DE
To find DE, we can solve this equation for DE. Multiplying both sides by 5/12:
(5/12) * 30 = DE
DE = 12.5