Textbook Question
In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.
P₁ = (-3, 0), P₂ = (-2, -2)
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 27
In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.
a = 14 meters, b = 12 meters, c = 4 meters
Verified step by step guidance1
Identify the side lengths of the triangle: \(a = 14\) meters, \(b = 12\) meters, and \(c = 4\) meters.
Calculate the semi-perimeter \(s\) of the triangle using the formula: \(s = \frac{a + b + c}{2}\).
Substitute the values of \(a\), \(b\), and \(c\) into the semi-perimeter formula to find \(s\).
Apply Heron's formula to find the area \(A\) of the triangle: \(A = \sqrt{s(s - a)(s - b)(s - c)}\).
Substitute the values of \(s\), \(a\), \(b\), and \(c\) into Heron's formula and simplify under the square root to find the area before rounding.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 area is found by √[s(s - a)(s - b)(s - c)]. This method avoids needing the height or angles.
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Quadratic Formula
Semi-perimeter of a Triangle
The semi-perimeter is half the perimeter of a triangle, calculated as s = (a + b + c) / 2. It is a key intermediate value in Heron's formula, simplifying the area calculation by incorporating all three side lengths into one term.
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5:3530-60-90 Triangles
Rounding and Units in Measurement
After calculating the area, it is important to round the result to the nearest whole number as specified. Additionally, the units of area are the square of the length units (e.g., square meters), reflecting the two-dimensional nature of area.
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06:11Introduction to the Unit Circle
Related Practice
Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
5v
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Textbook Question
In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.
w - v
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Textbook Question
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal. v = 2i - 2j, w = -i + j
Textbook Question
In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 7, b = 28, A = 12°
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Textbook Question
In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 30, b = 40, A = 20°