Answer: approximately 4.1307.
Explanation:
In triangle CDE, we are given the following information:
1. Angle C is twice angle D, which we can represent as: m∠C = 2 * m∠D.
2. Angle E is three times angle D, which we can represent as: m∠E = 3 * m∠D.
3. CD = 16.
To find the length of DE, we need to use the angle-angle-side (AAS) similarity criterion.
1. By angle-angle similarity, we know that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
2. Since m∠C = 2 * m∠D and m∠E = 3 * m∠D, we can conclude that △CDE is similar to △CED.
3. By side-angle-side (SAS) similarity, we know that if two pairs of corresponding sides are proportional and the included angles are congruent, then the triangles are similar.
4. In this case, CD corresponds to CE in the similar triangles, and we know that CD = 16.
5. Therefore, we can write the proportion: CD/CE = DE/CD.
6. Substitute the values we have: 16/CE = DE/16.
7. Cross-multiply to solve for DE: DE = (16 * DE) / CE.
8. Since △CDE and △CED are similar, we can equate the ratios: CD/CE = CE/DE.
9. Substitute the values we have: 16/CE = CE/DE.
10. Cross-multiply to solve for DE: DE = (CE * DE) / 16.
11. Simplify the equation: DE^2 = CE^2 / 16.
12. Take the square root of both sides to isolate DE: DE = √(CE^2 / 16).
13. Simplify further: DE = CE / 4.
14. From the given information, we know that m∠E = 3 * m∠D. Since the sum of angles in a triangle is 180 degrees, we have:
m∠C + m∠D + m∠E = 180.
2m∠D + m∠D + 3m∠D = 180.
6m∠D = 180.
m∠D = 30 degrees.
15. Now, we can find the measure of m∠E: m∠E = 3 * m∠D = 3 * 30 = 90 degrees.
16. Since m∠E is a right angle, we have a right triangle, and CD is the hypotenuse. By the Pythagorean theorem, we can find CE:
CE^2 = CD^2 - DE^2.
16^2 = CE^2 - (CE / 4)^2.
256 = CE^2 - (CE^2 / 16).
256 = 15CE^2 / 16.
CE^2 = (256 * 16) / 15.
CE^2 = 273.0667.
CE ≈ √273.0667.
CE ≈ 16.523.
17. Finally, we can substitute the value of CE back into the equation for DE: DE = CE / 4.
DE ≈ 16.523 / 4.
DE ≈ 4.1307.
Therefore, the length of DE is approximately 4.1307.