/*! 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 46 Show that the points form the ve... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 46

Show that the points form the vertices of the indicated polygon. Isosceles triangle: (2,3),(4,9),(-2,7)

Short Answer

Expert verified
Yes, the given points form the vertices of an isosceles triangle as two sides are of equal length.

Step by step solution

01

Calculation of distance between point (2,3) and (4,9)

Use the distance formula to find the length: \( \sqrt{(4-2)^2 + (9-3)^2} = \sqrt{2^2 + 6^2} = \sqrt{40} \)
02

Calculation of distance between point (2,3) and (-2,7)

Use the distance formula to find the length: \( \sqrt{(-2-2)^2 + (7-3)^2} = \sqrt{-4^2 + 4^2} = \sqrt{32} \)
03

Calculation of distance between point (4,9) and (-2,7)

Use the distance formula to find the length: \( \sqrt{(-2-4)^2 + (7-9)^2} = \sqrt{-6^2 + (-2)^2} = \sqrt{40} \)
04

Concluding the nature of the triangle

Compare the calculated lengths. If two of them are equal, the points form an isosceles triangle. In this case, the distances between the first and the third points are equal, hence the points form an isosceles triangle.

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.

Understanding the Isosceles Triangle
An isosceles triangle is a special kind of triangle where two of its sides are of equal length. This unique property makes it distinct from other types of triangles. The third side can be the longest or shortest, but two sides are always equally long.
When working with an isosceles triangle in geometry, one of the key insights is recognizing these equal sides. Not only does it help in classifying the triangle, but it also assists in solving geometric problems including finding missing angles or determining symmetry.
  • If you ever get a set of coordinates like (2,3), (4,9), and (-2,7) and need to figure out if they form an isosceles triangle, comparing the lengths between these points is crucial.
  • Always remember that in an isosceles triangle, because two sides are the same, the angles opposite those sides are also equal.
This property can be helpful in various geometry problems or proofs.
Exploring Coordinate Geometry
Coordinate geometry, also known as analytic geometry, merges algebra with geometry using a coordinate plane. It allows us to describe geometric figures numerically and solve geometric problems through algebraic processes.
In coordinate geometry, we use the coordinate system with the x-axis and y-axis to plot points and lines. This system helps us analyze shapes, sizes, and distances between points on the plane.
  • The points from our exercise are plotted as (2,3), (4,9), and (-2,7). To determine the type of triangle they form, we utilize the coordinate system effectively.
  • Using coordinates, you can calculate areas, perimeters, or angles of various shapes, but more importantly, you can visualize the problems at hand.
This tool is invaluable because it turns visual problems into numerical ones that can be easily tackled with algebraic methods.
Performing Distance Calculation
Distance calculation in coordinate geometry refers to finding the actual 'as-the-crow-flies' distance between two points. The distance formula is essentially a practical application of the Pythagorean theorem.
The formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where
  • \( (x_1, y_1) \) represent the coordinates of the first point, and
  • \( (x_2, y_2) \) represent the coordinates of the second point.
This formula calculates the linear distance between two coordinates on a plane by forming a right triangle and applying the theorem.
In the isosceles triangle exercise, distances calculated were:
  • Between (2,3) and (4,9) was \( \sqrt{40} \)
  • Between (2,3) and (-2,7) was \( \sqrt{32} \)
  • Between (4,9) and (-2,7) was \( \sqrt{40} \)
Recognizing these calculations as not just numbers, but segments of the triangle, allows us to conclude something about the shape they form. Two equal lengths indicate the isosceles nature.

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 Hooke's Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. The coiled spring of a toy supports the weight of a child. The spring is compressed a distance of 1.9 inches by the weight of a 25 -pound child. The toy will not work properly if its spring is compressed more than 3 inches. What is the weight of the heaviest child who should be allowed to use the toy?

Your wage is \(\$ 10.00\) per hour plus \(\$ 0.75\) for each unit produced per hour. So, your hourly wage \(y\) in terms of the number of units produced \(x\) is \(y=10+0.75 x\) (a) Find the inverse function. What does each variable represent in the inverse function? (b) Determine the number of units produced when your hourly wage is \(\$ 24.25\).

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{x-3}{x+2} $$

The mathematical model \(y=\frac{\kappa}{x}\) is an example of _____ variation.

Find a mathematical model for the verbal statement. \(V\) varies directly as the cube of \(e\).
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics 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.