Question 1179470: Sam owns a triangular piece of land on which the taxcollector wishes to determine the correctproperty tax. Sam tells the collector that “the firstside lies on a straight section of road and the secondside is a stone wall. The wall meets the road at a24-degree angle. The third side of the property isformed by a 180-foot-long fence, which meets the wallat a point that is 340 feet from the corner wherethe wall meets the road.” After a little thought,the tax collector realizes that Sam’s descriptionof hisproperty is ambiguous, because there are still twopossible lengths for the first side. By means ofa cleardiagram, explain this situation, and calculate thetwo possible areas, to the nearest square foot.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's a diagram that explains the ambiguous case and the calculations for the two possible areas of Sam's triangular piece of land.
**Diagram:**
[asy]
unitsize(0.8 cm);
pair A, B, C, D;
A = (0,5);
B = (0,0);
C = (8,0);
D = (8,3);
draw(A--B--C--cycle);
draw(C--D);
label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, NE);
label("$24^\circ$", B + (0.8,0.2));
label("$180$", (A + C)/2, NE);
label("$340$", (C + D)/2, NE);
[/asy]
Let:
* AB be the first side (road)
* BC be the second side (wall)
* AC be the third side (fence)
* D be the point where the fence meets the wall
We are given:
* Angle ABC = 24°
* AC = 180 feet
* CD = 340 feet
**Ambiguity:**
The ambiguity arises because we are not given which side of the wall the fence lies on. This leads to two possible triangles:
* **Triangle ABC:** The fence lies on the same side of the wall as the road.
* **Triangle ABD:** The fence lies on the opposite side of the wall as the road.
**Calculations:**
**Triangle ABC:**
1. **Find angle ACB:**
Using the Law of Sines in triangle ACD:
```
sin(∠ACD) / AC = sin(∠CAD) / CD
sin(∠ACD) / 180 = sin(24°) / 340
sin(∠ACD) ≈ 0.214
∠ACD ≈ 12.4°
```
Since ∠ACD and ∠ACB are supplementary angles:
```
∠ACB ≈ 180° - 12.4° = 167.6°
```
2. **Find angle BAC:**
```
∠BAC ≈ 180° - ∠ABC - ∠ACB
∠BAC ≈ 180° - 24° - 167.6° = -11.6°
```
Since angles cannot be negative, this triangle is not possible.
**Triangle ABD:**
1. **Find angle ADB:**
Using the Law of Sines in triangle ACD:
```
sin(∠ADB) / AC = sin(∠CAD) / CD
sin(∠ADB) / 180 = sin(24°) / 340
sin(∠ADB) ≈ 0.214
∠ADB ≈ 12.4°
```
2. **Find angle BAD:**
```
∠BAD ≈ 180° - ∠ABD - ∠ADB
∠BAD ≈ 180° - 24° - 12.4° = 143.6°
```
3. **Find AD:**
Using the Law of Sines in triangle ABD:
```
AD / sin(∠ABD) = BD / sin(∠BAD)
AD / sin(24°) = 340 / sin(143.6°)
AD ≈ 224.3 \text{ feet}
```
4. **Find AB:**
Using the Law of Cosines in triangle ABD:
```
AB^2 = AD^2 + BD^2 - 2 * AD * BD * cos(∠ADB)
AB^2 ≈ 224.3^2 + 340^2 - 2 * 224.3 * 340 * cos(12.4°)
AB ≈ 129.4 \text{ feet}
```
5. **Calculate the area:**
Using Heron's formula with sides AB, BD, and AD:
```
s = (AB + BD + AD) / 2 ≈ (129.4 + 340 + 224.3) / 2 ≈ 346.85
Area ≈ √[s(s - AB)(s - BD)(s - AD)]
Area ≈ √[346.85(346.85 - 129.4)(346.85 - 340)(346.85 - 224.3)]
Area ≈ 14595 \text{ square feet}
```
**Conclusion:**
The two possible lengths for the first side are approximately 129.4 feet. The area of the triangular piece of land is approximately 14595 square feet.
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