You are given triangle ABC. AB=13,BC=15, and CA=14. Point D is on segment BC and CD=6. Point E is on segment BC such that CE>CD and ∠BAE=∠CAD. What is the length of BE?
Do you know the difference between geography and geometry?
I'm sorry, should be geometry (math)
To find the length of BE, we need to use the properties of similar triangles.
1. First, let's draw triangle ABC with the given measurements:
- AB = 13
- BC = 15
- CA = 14
2. Next, mark point D on segment BC such that CD = 6.
3. Now, we need to find point E on segment BC such that CE > CD and ∠BAE = ∠CAD. This means that triangle BAE and triangle CAD are similar.
4. Since triangle BAE and triangle CAD are similar, we can write the following proportion:
AE / AD = AB / AC
5. Plugging in the known values:
AE / 6 = 13 / 14
6. To solve for AE, cross-multiply and solve the equation:
AE = (13 / 14) * 6
AE ≈ 5.571
7. Now, we have the length of AE. To find BE, we need to subtract DE from AE:
BE = AE - DE
8. To find DE, we can use the fact that CD = 6 and CE = 15 - 6 = 9.
9. Applying the Pythagorean theorem to triangle CDE, we have:
CD^2 + DE^2 = CE^2
6^2 + DE^2 = 9^2
DE^2 = 9^2 - 6^2
DE^2 = 81 - 36
DE^2 = 45
DE ≈ √45
DE ≈ 6.708
10. Finally, we can substitute the values of AE and DE into the equation for BE:
BE = 5.571 - 6.708
BE ≈ -1.137
The length of BE is approximately -1.137 units.