Textbook Question
Solve each triangle. See Examples 2 and 3.
B = 74.8°, a = 8.92 in., c = 6.43 in.
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7. Non-Right Triangles
Law of Cosines
Problem 63
Textbook Question
Find the exact area of each triangle using the formula 𝓐 = ½ bh, and then verify that Heron's formula gives the same result.
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Verified step by step guidance1
Identify the base (b) and height (h) of the triangle from the given image or problem statement. The base is one side of the triangle, and the height is the perpendicular distance from the opposite vertex to this base.
Use the formula for the area of a triangle: \(\mathcal{A} = \frac{1}{2} b h\). Substitute the values of the base and height into this formula to express the area.
Next, find the lengths of all three sides of the triangle. If not given, use the Pythagorean theorem or trigonometric ratios to calculate any missing side lengths.
Calculate the semi-perimeter \(s\) of the triangle using the formula \(s = \frac{a + b + c}{2}\), where \(a\), \(b\), and \(c\) are the lengths of the three sides.
Apply Heron's formula for the area: \(\mathcal{A} = \sqrt{s(s - a)(s - b)(s - c)}\). Substitute the side lengths and semi-perimeter into this formula to verify that the area matches the one found using \(\frac{1}{2} b h\).
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Area of a Triangle Using Base and Height
The area of a triangle can be calculated using the formula 𝓐 = ½ × base × height, where the base is any side of the triangle and the height is the perpendicular distance from the opposite vertex to that base. This method requires knowing or determining the height explicitly.
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Heron's Formula
Heron's formula calculates the area of a triangle when the lengths of all three sides are known. It uses the semi-perimeter s = (a + b + c)/2 and the formula 𝓐 = √[s(s - a)(s - b)(s - c)], allowing area calculation without needing the height.
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Verification of Area Using Two Methods
Verifying the area by both the base-height formula and Heron's formula ensures accuracy and deepens understanding. It involves calculating the area using both methods and confirming that the results match, demonstrating consistency between geometric and algebraic approaches.
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