/*! 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 39 WRITING Explain the difference b... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 39

WRITING Explain the difference between the circum center and the in center of a triangle.

Short Answer

Expert verified
The circumcenter of a triangle is the intersection of the perpendicular bisectors of the sides and the center of the circumcircle, which can be located inside, outside, or on the triangle. The incenter of a triangle, on the other hand, is the intersection of the angle bisectors and the center of the incircle, and is always located within the triangle.

Step by step solution

01

Define the circumcenter

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is the center of the circumscribing circle (circumcircle) that passes through each of the triangle's vertices. The circumcenter can be located inside, outside, or on the triangle based on the type of triangle (acute, obtuse, or right respectively).
02

Define the incenter

The incenter of a triangle is the point where the angle bisectors of the triangle meet. It is the center of the inscribed circle (incircle) that is tangent to each of the triangle's sides. The incenter is always located inside the triangle, regardless of the triangle's shape.
03

Express the difference

While both the circumcenter and incenter are associated with circles around or within a triangle, they differ in their construction and location. The circumcenter uses the perpendicular bisectors of the sides while the incenter uses the angle bisectors. The circumcenter's location might vary with the shape of the triangle, but the incenter is always inside the 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.

Circumcenter
The circumcenter is a fascinating point within a triangle, defined by its geometric properties. It is the spot where all three perpendicular bisectors of a triangle's sides converge.
These perpendicular bisectors are lines that intersect a side at a right angle and at its midpoint, essentially cutting it perfectly in two.
  • The circumcenter serves as the center of a circumcircle, a circle that passes through each vertex of the triangle.
  • This circumcircle exemplifies how every triangle's vertices lie equidistant from the circumcenter.
A key aspect of the circumcenter is how its location changes based on the type of triangle:
  • In an **acute triangle**, the circumcenter falls inside.
  • For an **obtuse triangle**, it is found outside.
  • In a **right triangle**, it resides directly on the hypotenuse.
This knowledge helps us understand relationships within various types of triangles.
Incenter
The incenter is another crucial triangle center, unique in its construction and location. This point is determined by where all the angle bisectors of a triangle meet.
An angle bisector divides an angle into two equal smaller angles, and each triangle has three such bisectors.
  • The incenter is the heart of the incircle, a circle perfectly tangent to each side of the triangle, hugging it inwardly.
  • This center is equidistant from all sides of the triangle, proving its consistent and special placement.
Unlike the circumcenter, the incenter is always nestled within the bounds of the triangle. Regardless of whether the triangle is acute, obtuse, or right, the incenter maintains this position. It鈥檚 this predictability that makes it a vital part of triangle geometry.
Perpendicular Bisectors
Perpendicular bisectors are lines playing a major role in the creation of the circumcenter. Each triangle side has its own perpendicular bisector, a segment that:
  • Splits a side into two equal halves
  • Runs at a 90-degree angle (perpendicular)
By definition, the intersection of these three bisectors in any triangle will always find the circumcenter of the triangle.
This intersection ensures all points of the triangle are equidistant to the circumcenter. Perpendicular bisectors highlight balance and uniformity by evenly dividing triangle segments.
Angle Bisectors
Angle bisectors, on the other hand, are all about angles as their name suggests. Each angle of the triangle boasts its own bisector, an imaginary line extending from a vertex that splits the angle into two equal parts.
The intersection of these angle bisectors pinpoints the incenter, which confirms it as:
  • Equidistant from all triangle sides
  • A constant presence inside the triangle
These bisectors emphasize symmetry and equivalence in the triangle, ensuring that the incenter remains eternally nestled within any triangle, whether obtuse, acute, or right. Mastery of angle and perpendicular bisectors deepens the understanding of triangle geometry and its intrinsic relationships.

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

Prove the Triangle Inequality Theorem (Theorem 6.11). Given \(\triangle \mathrm{ABC}\) Prove \(A B \quad B C>A C, A C \quad B C>A B\) and \(AB \quad AC>BC\)

In Exercises \(15-18\) , \(\square\)nd the coordinates of the centroid of the triangle with the given vertices. ( See Example 2 ) $$\mathrm{A}(2,3), \mathrm{B}(8,1), \mathrm{C}(5,7)$$

In Exercises 31鈥�36, complete the statement with always, sometimes, or never. Explain your reasoning. The centroid is _______ on the triangle.

Exercises \(23-26,\) draw the indicated triangle and \(\square\)nd its centroid and orthocenter. acute isosceles triangle

MODELING WITH MATHEMATICS are placing a fountain in a triangular koi pond. You want the fountain to be the same distance from each edge of the pond. Where should you place the fountain? Explain your reasoning. Use a sketch to support your answer.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

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.