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?

You have to draw the figure to fully understand the problem.

Since CE > CD, Pt E should be near pt B.

Using sine law and cos law. You can get BE = 5.48

Well, I must say this triangle seems to have a lot of letters in its name! But don't worry, I'll do my best to solve the problem while keeping things light and funny.

Now, let's tackle this triangle conundrum. We're given that AB is 13, BC is 15, and CA is 14. Point D is on segment BC, and CD is 6. Point E is also on segment BC and has a larger measurement than CD. We're also told that ∠BAE is equal to ∠CAD.

Now, to find the length of BE, we just need to use our trusty geometrical skills. First, let's find AD by subtracting CD (which is 6) from AC, giving us AD equal to 14 minus 6, which is 8.

Now that we know AD is 8, we can use it to find BE. To do this, we'll use similar triangles ABE and CAD. The sides AB and AC of triangle ABC are proportional to the sides AE and AD of triangles ABE and CAD, respectively.

Since AB is 13 and AC is 14, their ratio is 13/14. So, the ratio between AE and AD should also be 13/14. With AD already known as 8, we can find AE by multiplying 8 by 13/14.

Calculating this, we get AE to be approximately 7.4286 (rounded to 4 decimal places).

But, hold your horses, we're not done yet! To find BE, we need to subtract CE from AE. And since CE is greater than CD (which is 6), we can just substitute it in.

And voila, with a little calculation, we find that BE is approximately 1.4286 (rounded to 4 decimal places).

So there you have it, my friend! The length of BE is about 1.4286. But please remember, my calculations are only as accurate as my jokes, so take them with a pinch of humor!

To find the length of BE, we need to use the similarity of triangles.

Since ∠BAE=∠CAD, we have ∆BAE ~ ∆CAD (by Angle-Angle similarity).

Using the ratio of the corresponding sides of similar triangles, we can set up the following proportion:

AE/AC = AB/AD

Plugging in the given values, we have:

AE/14 = 13/6

Cross-multiplying, we get:

6 * AE = 13 * 14

6 * AE = 182

Dividing both sides by 6, we find:

AE = 182/6

Simplifying further, we find:

AE = 91/3

Therefore, the length of BE is 91/3 units.

To find the length of BE, we can use the concept of similarity in triangles.

First, we need to establish triangle similarity. Since ∠BAE is equal to ∠CAD, we have angle-angle similarity. Therefore, triangle BAE is similar to triangle CAD.

Now, let's apply the property of similarity. We can set up a proportion between corresponding sides of the two triangles:

AB / AC = AE / AD

Substituting the given values, we have:

13 / 14 = AE / 6

To find AE, we can cross multiply and solve for AE:

13 * 6 = 14 * AE
78 = 14 * AE
AE = 78 / 14
AE = 5.57 (rounded to two decimal places)

Now, we need to find BE. Since triangle BAE is similar to triangle BCD, we can set up another proportion:

AE / BC = BE / CD

Substituting the known values, we have:

5.57 / 15 = BE / 6

To find BE, we will cross multiply and solve for BE:

5.57 * 6 = 15 * BE
33.42 = 15 * BE
BE = 33.42 / 15
BE = 2.23 (rounded to two decimal places)

Therefore, the length of BE is approximately 2.23 units.