/*! 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 37 Using Heron's Area Formula use... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 37

Using Heron's Area Formula use Heron's Area Formula to find the area of the triangle. $$ a=8, \quad b=12, \quad c=17 $$

Short Answer

Expert verified
The area of the triangle is approximately 37.93 square units.

Step by step solution

01

Calculate the semi-perimeter of the triangle.

The semi-perimeter of a triangle with side lengths \(a\), \(b\), and \(c\) is calculated as \(s = (a+b+c)/2\). Here, \(s = (8+12+17)/2 = 18.5\).
02

Apply Heron's Formula

Using the semi-perimeter, calculate the area of the triangle using Heron's Formula: \[ Area = \sqrt{s(s-a)(s-b)(s-c)} \] Substituting the given values, we get \( Area = \sqrt{18.5(18.5-8)(18.5-12)(18.5-17)}\) which then equals approximately 37.93 square units when simplified.

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.

Semi-Perimeter of a Triangle
Understanding the semi-perimeter of a triangle is essential in solving various triangle-related problems, including using Heron's Area Formula. The semi-perimeter, often denoted by the symbol \(s\), is simply half of the triangle's perimeter. To find it, one must first determine the sum of all the side lengths and then divide by two.

For a triangle with sides of lengths \(a\), \(b\), and \(c\), you can find the semi-perimeter using the formula:
\[s = \frac{a + b + c}{2}\]
Once you have this value, it is used in several mathematical formulas related to triangles, such as Heron's Area Formula, which cleverly avoids the need for calculating angles or heights.
Triangle Area Calculation
The area of a triangle is a fundamental concept in geometry, representing the amount of space enclosed by the triangle. There are multiple ways to calculate it, including using base and height, trigonometry, or the method we're focusing on, Heron's Area Formula.

Heron's formula is particularly useful when you know the side lengths of the triangle but not the height. It is given by:
\[Area = \sqrt{s(s-a)(s-b)(s-c)}\]

Applying Heron's Formula

After finding the semi-perimeter \(s\) as outlined in the previous section, you plug in the values of \(s\), \(a\), \(b\), and \(c\) into Heron's formula to calculate the area. Poetically, it ties together all three sides of the triangle in a symphony of multiplication and square roots, elegantly producing the area without further complication.
Precalculus Mathematics
Precalculus mathematics serves as the bridge between the concrete world of algebra and the abstract territory of calculus. It involves the study of functions, analytical geometry, and other mathematical concepts that prepare students for understanding the rate of change and the accumulation functions in calculus.

Heron's Area Formula, which we've discussed here, is part of precalculus mathematics. It embodies the beauty of precalculus by showing how principles of algebra and geometry come together to solve complex problems. As students advance through precalculus, they learn not only to solve problems but to understand how various formulas and theorems are derived, setting a strong foundation for calculus challenges.

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

Einding the Component Form of a Vector In Exercises \(75-78\) , find the component form of the sum of u and v with direction angles \(\theta_{\text { u }}\) and \(\theta_{v}\) . $$\begin{array}{ll}{\text { Magnitude }} & {\text { Angle }} \\\ {\|\mathbf{u}\|=5} & {\theta_{\mathrm{u}}=0^{\circ}} \\ {\|\mathbf{v}\|=5} & {\theta_{\mathrm{v}}=90^{\circ}}\end{array}$$

Think About It What can be said about the vectors \(\mathbf{u}\) and \(\mathbf{v}\) under each condition? (a) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{u}\) . (b) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{0}\) .

A commercial jet is flying from Miami to Seattle. The jet's velocity with respect to the air is 580 mile. per hour, and its bearing is \(332^{\circ} .\) The wind, at the altitude of the plane, is blowing from the southwest with a velocity of 60 miles per hour. (a) Draw a figure that gives a visual representation of the situation. (b) Write the velocity of the wind as a vector in component form. (c) Write the velocity of the jet relative to the air in component form. (d) What is the speed of the jet with respect to the ground? (e) What is the true direction of the jet?

Solving an Equation In Exercises \(97-104,\) use the formula on page 446 to find all solutions of the equation and represent the solutions graphically. $$x^{5}+243=0$$

Work A ski patroller pulls a rescue toboggan across a flat snow surface by exerting a force of 35 pounds on a handle that makes an angle of \(22^{\circ}\) with the horizontal. Find the work done in pulling the toboggan 200 feet.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.