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Chapter 9: Problem 8

Heron's Formula is used to find the area of (a) ASA (b) SAS (c) SSS (d) AAS

Short Answer

Expert verified
The answer is (c) SSS.

Step by step solution

01

Understand the problem

The exercise requires identifying which triangle configuration Heron's Formula is used to find the area of. The options provided are ASA (Angle-Side-Angle), SAS (Side-Angle-Side), SSS (Side-Side-Side), and AAS (Angle-Angle-Side).
02

Recall Heron's Formula

Heron's Formula states that the area of a triangle can be calculated using the lengths of all three sides. It is given by \ \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] where \ s = \frac{a+b+c}{2} \ and \ a, b, c \ are the lengths of the sides of the triangle.
03

Match the formula with the triangle type

Since Heron's Formula uses the lengths of all three sides of the triangle to calculate the area, it corresponds to the triangle configuration where all three sides are known, which is the SSS (Side-Side-Side) configuration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle Area Calculation
Calculating the area of a triangle is a fundamental aspect of geometry. Various methods exist depending on the information available about the triangle. For example, some common methods include:
  • Using base and height: Area = 0.5 脳 base 脳 height
  • Trigonometrical formulas: When you have one side and two angles (ASA, AAS), or two sides and an included angle (SAS)
  • Heron's Formula: Particularly useful when you know the lengths of all three sides (SSS)
The triangle area calculation becomes much easier when knowing the right formula to use, and Heron's formula comes in handy for SSS triangles. Let鈥檚 delve deeper into this special case.
SSS Triangle
When dealing with the SSS (Side-Side-Side) triangle configuration, all three sides are known. Heron's Formula is unique as it works best for this scenario. Here鈥檚 why:
  • Uses all three side lengths to find the area
  • Does not require any angles, which makes it versatile for many different kinds of triangles
To apply Heron's formula, you need three steps:1. Calculate the semi-perimeter (\[ s = \frac{a + b + c}{2} \])2. Substitute the semi-perimeter and the side lengths into the Heron鈥檚 Formula (\[ A = \sqrt{s(s-a)(s-b)(s-c)} \])3. Solve to find the area of the triangleThese steps are straightforward, allowing anyone to find the area without needing advanced geometric tools.
Mathematical Formulas
Heron's Formula stands out as an elegant geometrical solution. Here鈥檚 a closer look at the formula:
  • Semi-perimeter (\[ s \]) is half the perimeter of the triangle
  • The formula: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] utilizes \[ s \] and the side lengths (\[ a, b, c \])
This allows the calculation of the triangle's area using pure numerical values of the sides.Consider the semi-perimeter first, it鈥檚 calculated as:\below seamiperimeter \[ s = \frac{a + b + c}{2} \]nenskapewising the complete formula, we substitute \[s, a, b, \] and \[ c:\] \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] Solving this gives you the triangle's area, taking the mystery out of calculating the area from side lengths alone.

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Most popular questions from this chapter

Adam must fly home to St. Louis from a business meeting in Oklahoma City. One flight option flies directly to St. Louis, a distance of about 461.1 miles. A second flight option flies first to Kansas City and then connects to St. Louis. The bearing from Oklahoma City to Kansas City is N29.6 \({ }^{\circ} \mathrm{E}\), and the bearing from Oklahoma City to St. Louis is N57.7 \(^{\circ}\) E. The bearing from St. Louis to Oklahoma City is \(S 57.7^{\circ}\) W, and the bearing from St. Louis to Kansas City is \(\mathrm{N} 79.4^{\circ} \mathrm{W}\). How many more frequent flyer miles will Adam receive if he takes the connecting flight rather than the direct flight?

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Tuning Fork The end of a tuning fork moves in simple harmonic motion described by the function \(d(t)=a \sin (\omega t)\) If a tuning fork for the note A above middle \(\mathrm{C}\) on an even-tempered scale \(\left(A_{4},\right.\) the tone by which an orchestra tunes itself) has a frequency of 440 hertz (cycles per second), find \(\omega\). If the maximum displacement of the end of the tuning fork is 0.01 millimeter, find a function that describes the movement of the tuning fork.
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