abcd is a square, and point e lies along side ab so that ae is three times as long as eb. To the nearest degree, what is measure of angle ecb?
To find the measure of angle ECB, we need to use the information given about the square ABCD and point E.
First, let's draw a diagram to visualize the problem:
```
A________B
| |
| |
e | |
| |
|________|
C D
```
We know that ABCD is a square. Let's label the side length of the square as 's'.
Now, the problem states that point E lies along side AB, and AE is three times as long as EB. So, let's label EB as 'x'. Thus, AE will be '3x' (three times the length of EB).
Since ABCD is a square, all sides are equal in length. Therefore, AB = BC = CD = DA = s.
Furthermore, we also know that AE + EB = AB (the total length of side AB). So, we can write the equation as:
3x + x = s
Since AE + EB = AB and AE is three times as long as EB, we can also write the equation as:
3x + 1x = s
4x = s
Now, let's find the value of x by substituting the side length of the square (s) in place of 's':
4x = s
4x = s = s
4x = s
Now, to find the measure of angle ECB, we can use trigonometry. In a right-angled triangle, the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side. In this case, angle ECB is the angle between sides EB and BC. The length of the opposite side is 'x', and the length of the adjacent side is 's'. Therefore, the tangent of angle ECB is given by:
tan(ECB) = x / s
To find the measure of angle ECB, we need to take the arctangent (inverse tangent) of the ratio:
ECB = arctan(x / s)
Now, we can substitute the value of x = s / 4 to find the measure of angle ECB:
ECB = arctan((s / 4) / s)
ECB = arctan(1 / 4)
Using a calculator, we can find the approximate value of this angle to the nearest degree.