Triangle ACE is a circumscribed triangle. Points B, D and F are the points of tangency. If AC= 16 and AE is 9 cm and CE = 15 cm . Find AD, FE and CD
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To find AD, FE, and CD, we can use properties of tangents to a circle.
1. AD is the tangent from point A to the circle. Since AB and AE are tangents, we can use the Tangent-Secant Theorem: (AD)^2 = AC * AE.
Substituting the given values, (AD)^2 = 16 * 9 cm^2. Calculating this gives: (AD)^2 = 144 cm^2. Therefore, AD = √144 cm = 12 cm.
2. FE is the tangent from point E to the circle. Since CF and CE are tangents, we can again use the Tangent-Secant Theorem: (FE)^2 = FC * CE.
To find FC, we can subtract FD from CD: CD = AC - AD. Substituting the given values, CD = 16 cm - 12 cm = 4 cm.
Now, using the Tangent-Secant Theorem, (FE)^2 = FC * 15 cm. Substituting the value of FC, (FE)^2 = 4 cm * 15 cm. Calculating this gives: (FE)^2 = 60 cm^2. Therefore, FE = √60 cm = 2√15 cm.
3. CD is the tangent from point C to the circle. Again, using the Tangent-Secant Theorem, (CD)^2 = CF * CA.
Substituting the given values, (CD)^2 = FC * 16 cm. Rearranging the equation, FC = (CD)^2 / 16 cm. Substituting the value of CD, we get FC = (4 cm)^2 / 16 cm = 1 cm.
Now, using the given values, (CD)^2 = 1 cm * 16 cm. Calculating this gives: (CD)^2 = 16 cm^2. Therefore, CD = √16 cm = 4 cm.
Therefore, AD = 12 cm, FE = 2√15 cm, and CD = 4 cm.
To find AD, FE, and CD, we can use the properties of a circumscribed triangle.
First, let's draw the figure:
```
A
/ \
/ \
B ----- C
| |
| F |
| |
D ----- E
```
In a circumscribed triangle, the radius of the circumcircle is the same length for all three sides. Let's label this radius as r.
We can start by finding the radius (r) using the formula:
```
r = (AC * AE * CE) / (4 * Area)
```
To find the area of triangle ACE, we can use Heron's formula:
```
s = (AC + AE + CE) / 2
Area = √(s * (s - AC) * (s - AE) * (s - CE))
```
Substituting the given values:
```
AC = 16
AE = 9
CE = 15
```
We can calculate s:
```
s = (16 + 9 + 15) / 2 = 20
```
Then, calculate the area:
```
Area = √(20 * (20 - 16) * (20 - 9) * (20 - 15)) = √(20 * 4 * 11 * 5) = √(4400) ≈ 66.33
```
Now, substitute the values into the radius formula:
```
r = (16 * 9 * 15) / (4 * 66.33) ≈ 6.12
```
Next, we can calculate AD, FE, and CD using the Pythagorean theorem.
AD:
In triangle ABD, AD is the hypotenuse. The other two sides are r (the radius) and AE.
```
AD^2 = AE^2 + r^2
AD^2 = 9^2 + (6.12)^2
AD^2 = 81 + 37.5344
AD^2 ≈ 118.5344
AD ≈ √(118.5344) ≈ 10.89
```
FE:
In triangle CEF, FE is the hypotenuse. The other two sides are r (the radius) and CE.
```
FE^2 = CE^2 + r^2
FE^2 = 15^2 + (6.12)^2
FE^2 = 225 + 37.5344
FE^2 ≈ 262.5344
FE ≈ √(262.5344) ≈ 16.19
```
CD:
In triangle CFD, CD is the hypotenuse. The other two sides are r (the radius) and CF. Since CF is equal to CE (both radii), CD is equal to 2 * r.
```
CD = 2 * r = 2 * 6.12 = 12.24
```
Therefore, the approximate values for AD, FE, and CD are:
AD ≈ 10.89 cm
FE ≈ 16.19 cm
CD ≈ 12.24 cm