/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 73 In Exercises 71-74, find the are... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 73

In Exercises 71-74, find the area of the triangle. $$ a=11, b=18, c=10 $$

Short Answer

Expert verified
Use the given side lengths and Heron's formula to calculate the area of the triangle. The semi-perimeter s is first calculated, then substituted into the formula.

Step by step solution

01

Calculate semi-perimeter

First, calculate the semi-perimeter (s) of the triangle, which is half the sum of its side lengths. Using the given lengths a=11, b=18, c=10, the formula is: \( s = \frac{a + b + c}{2} \).
02

Apply Heron's formula

Next, use Heron's formula to calculate the area (A) of the triangle. The formula is: \( A = \sqrt{s(s-a)(s-b)(s-c)} \). Remember to carry out multiplication and division operations before addition and subtraction (according to the order of operations).
03

Solve for A.

Substitute the calculated values of s, a, b, and c into Heron's formula, then simplify as needed to find A.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Heron's Formula
Heron's formula is a powerful tool in geometry that calculates the area of a triangle when you know the lengths of all three sides. It is particularly useful for triangles where the height is not readily available or easy to determine. The formula is given as:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Where:
  • \( A \) is the area of the triangle
  • \( s \) is the semi-perimeter
  • \( a, b, \) and \( c \) are the lengths of the sides of the triangle
The beauty of Heron's formula is its simplicity and elegance, providing a direct way to find the area using just the side lengths. Keep in mind, care must be taken with mathematical accuracy during calculation, as small mistakes can lead to incorrect results.
Semi-Perimeter
Before you can use Heron's formula, you need to calculate the semi-perimeter of the triangle.The semi-perimeter \( s \) is half of the perimeter of the triangle. For a triangle with side lengths \( a, b, \) and \( c \), the semi-perimeter is calculated as:\[ s = \frac{a + b + c}{2} \]In our original exercise, substituting the side lengths \( a = 11, b = 18, \) and \( c = 10 \), we calculate:\[ s = \frac{11 + 18 + 10}{2} = 19.5 \]The semi-perimeter serves as a critical intermediate step, enabling us to seamlessly move forward and apply Heron's formula. It divides the perimeter evenly, simplifying the calculations when inserted into Heron's formula.
Order of Operations
Order of operations is crucial when calculating mathematical expressions. This is especially true in formulas like Heron's where multiple mathematical operations are combined. The order of operations consists of:
  • Parentheses
  • Exponents and roots
  • Multiplication and division
  • Addition and subtraction
Known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this rule ensures that calculations are carried out in the correct sequence. In Heron's formula, after calculating the semi-perimeter, multiplication and subtraction are performed within the radical operator (square root), aligning with these rules. Ensuring the correct order prevents miscalculations and aids in finding the accurate area of the triangle.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use a graphing utility to graph and identify \(r=2+k \sin \theta\) for \(k=0,1,2\), and 3 .

In Exercises 53-56, use a graphing utility to graph the polar equation and show that the indicated line is an asymptote of the graph. $$ \begin{array}{ccc} \text { Name of Graph } & \text { Polar Equation } & \text { Asymptote } \\ \text { Conchoid } & r=2-\sec \theta & x=-1 \end{array} $$

\(r=4+3 \cos \theta\)

In Exercises 79-82, find the exact value of the trigonometric function given that \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$ \sin (u-v) $$

In Exercises 79-82, find the exact value of the trigonometric function given that \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$ \cos (u+v) $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.