Textbook Question
In Exercises 17–22, find the angle between v and w. Round to the nearest tenth of a degree.v = 2i - j, w = 3i + 4j
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 16
In Exercises 13–16, find the area of the triangle having the given measurements. Round to the nearest square unit.a = 2 meters, b = 2 meters, c = 2 meters
Verified step by step guidance1
Recognize that the given triangle is an equilateral triangle since all sides are equal: \( a = b = c = 2 \) meters.
Use the formula for the area of an equilateral triangle: \( \text{Area} = \frac{\sqrt{3}}{4} a^2 \).
Substitute the side length \( a = 2 \) meters into the formula: \( \text{Area} = \frac{\sqrt{3}}{4} \times 2^2 \).
Simplify the expression: \( \text{Area} = \frac{\sqrt{3}}{4} \times 4 \).
Calculate the area and round to the nearest square unit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Triangle Area Formulas
The area of a triangle can be calculated using various formulas, depending on the information available. For triangles with known side lengths, Heron's formula is particularly useful. It states that the area can be found using the semi-perimeter and the lengths of the sides. For this triangle, since all sides are equal, the formula simplifies the calculation.
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Heron's Formula
Heron's formula allows for the calculation of the area of a triangle when the lengths of all three sides are known. It involves first calculating the semi-perimeter (s = (a + b + c) / 2) and then applying the formula: Area = √(s(s-a)(s-b)(s-c)). This method is particularly effective for non-right triangles, such as the equilateral triangle in this problem.
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Equilateral Triangle Properties
An equilateral triangle has all three sides of equal length and all angles measuring 60 degrees. This symmetry simplifies calculations, as the height can be derived from the side length. For an equilateral triangle with side length 'a', the area can also be calculated using the formula: Area = (√3/4) * a², which provides a quick way to find the area without using Heron's formula.
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Related Practice
Textbook Question
In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.B = 80°, C = 10°, a = 8
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Textbook Question
In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.a = 5, c = 2, B = 90°
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Textbook Question
In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.P₁ = (-1, 7), P₂ = (-7, -7)
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Textbook Question
In Exercises 9–16, letu = 2i - j, v = 3i + j, and w = i + 4j.Find each specified scalar.4(u ⋅ v)
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Textbook Question
In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.a = 5, b = 7, c = 10