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Magazine Chapter 7: Problem 16
Solve each of the following triangles. $$ a=832 \mathrm{ft}, b=623 \mathrm{ft}, c=345 \mathrm{ft} $$
Short Answer
Expert verified
The triangle exists. Its area is approximately 3291.6 sq ft, with angles \( A \approx 64.8^\circ \), \( B \approx 41.2^\circ \), \( C \approx 74^\circ \).
Step by step solution
01
Verify Triangle Validity
To determine if a triangle can exist with the given side lengths, use the triangle inequality theorem. According to the theorem: 1. The sum of the lengths of any two sides must be greater than the length of the third side.2. Check: - Is \( a + b > c \)? That is \( 832 + 623 > 345 \)? Yes, since 1455 > 345. - Is \( a + c > b \)? That is \( 832 + 345 > 623 \)? Yes, since 1177 > 623. - Is \( b + c > a \)? That is \( 623 + 345 > 832 \)? Yes, since 968 > 832.All conditions satisfy the theorem, hence, a triangle can exist.
02
Calculate the Triangle's Area Using Heron's Formula
We can find the area of the triangle using Heron's formula. First, calculate the semi-perimeter \( s \):\[s = \frac{a + b + c}{2} = \frac{832 + 623 + 345}{2} = 900 \, \text{ft}\]Apply Heron's formula for the area:\[K = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)} \]Substitute the values:\[K = \sqrt{900 \cdot (900 - 832) \cdot (900 - 623) \cdot (900 - 345)} \K = \sqrt{900 \cdot 68 \cdot 277 \cdot 555} \]Calculate and simplify to find the area.
03
Compute the Area
Continuing from Step 2, calculate the actual value of the expression:\[K = \sqrt{900 \cdot 68 \cdot 277 \cdot 555} \]This calculates to around:\[K \approx 10,839,600 \]Taking the square root gives:\[K \approx 3291.6 \, \text{sq ft}\]The area of the triangle is approximately 3291.6 square feet.
04
Determine Angles Using the Law of Cosines
The Law of Cosines helps find the angles. For angle \( A \), use:\[a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \]Substitute the given values and solve for \( \cos(A) \):\[832^2 = 623^2 + 345^2 - 2 \cdot 623 \cdot 345 \cdot \cos(A)\]Calculate and isolate \( \cos(A) \). Subsequently, find \( A \) using \( \cos^{-1}(A) \).
05
Calculate Specific Angle
Solve the expression:\[692224 = 388129 + 119025 - 2 \cdot 623 \cdot 345 \cdot \cos(A)\]Simplify and solve:\[692224 = 507154 - 429570 \cdot \cos(A)\]\[184070 = 429570 \cdot \cos(A)\]\[\cos(A) = \frac{184070}{429570}\]Calculate \( A \) using \( \cos^{-1}(A) \). Thus, \( A \approx 64.8^\circ \).
06
Repeat for Remaining Angles
Repeat the process described in Steps 4 and 5 for angles \( B \) and \( C \) using the Law of Cosines.For \( B \):\[623^2 = 832^2 + 345^2 - 2 \cdot 832 \cdot 345 \cdot \cos(B)\]For \( C \):\[345^2 = 832^2 + 623^2 - 2 \cdot 832 \cdot 623 \cdot \cos(C)\]Calculate each cosine and use \( \cos^{-1} \) to find \( B \approx 41.2^\circ \) and \( C \approx 74^\circ \).
07
Verify Angle Sum
Add angles \( A, B, C \) to verify the sum equals 180 degrees:\[A + B + C = 64.8^\circ + 41.2^\circ + 74^\circ = 180^\circ\]The sum is correct which confirms the angle measurements are accurate.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Triangle Inequality Theorem
The Triangle Inequality Theorem is a fundamental concept that helps us determine whether a set of three side lengths can construct a valid triangle. It's like a checklist to verify the possibility of forming a triangle:
- First Condition: The sum of any two sides must be greater than the third side.
- We check this condition three times by combining different side pairs.
- For Example: For sides a, b, and c, verify: - Is \(a + b > c\)? - Is \(a + c > b\)? - Is \(b + c > a\)?
- If all conditions are satisfied, it means a triangle can exist.
- We have side lengths a = 832, b = 623, and c = 345.
- Checking conditions one-by-one, all inequalities hold true.
- This shows that a triangle is possible with these dimensions.
Heron's Formula
Calculating the area of a triangle with given side lengths can be accomplished through Heron's Formula. This formula doesn't require any angles and is useful when you have the measurements of all sides.
- The Formula is: \( K = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)} \)
- First, determine the semi-perimeter \( s \) : \( s = \frac{a + b + c}{2} \)
- For Our Example: With sides a = 832 ft, b = 623 ft, and c = 345 ft, the semi-perimeter = 900 ft.
- Substituting these into Heron's formula provides the area.
Law of Cosines
The Law of Cosines is instrumental in finding unknown angles in a triangle, particularly when side lengths are known. This law helps extend the Pythagorean theorem to non-right triangles:
- Law of Cosines Formula: For any triangle with sides a, b, c and corresponding angles A, B, C:
- \(a^2 = b^2 + c^2 - 2bc \cdot \cos(A)\)
- Similarly applied for other angles:
- \(b^2 = a^2 + c^2 - 2ac \cdot \cos(B)\)
- \(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
- In Our Exercise: To find angle A using the side lengths:
- Plug values of sides into the equation to solve for \(\cos(A)\).
- This determines the measure of angles using a calculator for inverse cosines.
- Repeat for angles B and C when necessary.
- Ensure the sum of angles equals 180掳 to validate calculations.
