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Chapter 6: Problem 52

Construct an acute scalene triangle. Find the orthocenter, centroid, and circumcenter. What can you conclude about the three points of concurrency?

Short Answer

Expert verified
The orthocenter, centroid, and circumcenter of an acute scalene triangle are all found inside the triangle. This observation about the points of concurrency is specific to acute triangles.

Step by step solution

01

Construct an Acute Scalene Triangle

First off, draw a triangle with all angles less than 90 degrees and with all sides of different lengths. You can use a ruler and protractor to ensure accuracy. This will be the acute scalene triangle needed for the exercise.
02

Find the Orthocenter

To locate the orthocenter of the triangle, draw lines from each vertex of the triangle perpendicular to the opposite side (these are the altitudes of the triangle). The point where these lines intersect is the orthocenter of the triangle.
03

Find the Centroid

The medians of a triangle are the lines drawn from each vertex to the midpoint of the opposite side. Draw these lines. Where they intersect is the centroid of the triangle, which is also the center of gravity.
04

Find the Circumcenter

Perpendicular bisectors of a triangle are the lines passing through the midpoint of each side and perpendicular to that side. Draw these lines and the point where they intersect is the circumcenter, which is also the center of the circle that can be drawn around the triangle (circumcircle).
05

Observation about the points of concurrency

Notice that in an acute triangle, all three points (centroid, orthocenter, and circumcenter) are all located inside the triangle. This will not be the case for other types of triangles (e.g., obtuse, right triangles).

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Key Concepts

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

Acute Scalene Triangle
An acute scalene triangle is a type of triangle where each angle measures less than 90 degrees, and all its sides are of different lengths. This means it has neither right angles, like a right triangle, nor equal sides, like an equilateral triangle. To construct such a triangle, use a protractor to ensure all angles are acute, and a ruler to measure each side to make sure they are not equal. The distinguishing feature of an acute scalene triangle is that every pair of sides and angles are unique from each other. This uniqueness makes the exploration of its geometrical properties fascinating.
Orthocenter
The orthocenter of a triangle is the point where all three altitudes intersect. Altitudes are perpendicular lines drawn from the vertices to the opposite sides. In an acute scalene triangle, these altitudes all meet within the triangle. Here's how you can find it:
  • Draw a line from one vertex to the opposite side, ensuring it meets the side at a right angle.
  • Repeat this process for the other two vertices.
  • The point where all three lines cross is the orthocenter.
Finding the orthocenter involves understanding the right angle relationships within a triangle, emphasizing the unique nature of the triangle's interaction with perpendicular lines.
Centroid
A centroid of a triangle is the point formed where all three medians intersect. A median is a line drawn from a vertex to the midpoint of the opposite side. This point acts as the triangle's center of gravity, often called its balance point. To locate the centroid in an acute scalene triangle:
  • Find the midpoint of each side of the triangle.
  • Draw a line from each vertex to the opposite midpoint (these are the medians).
  • The intersection point of these medians is the centroid.
The centroid has the property of dividing each median into two segments where the longer segment is twice as long as the shorter segment from the vertex.
Circumcenter
The circumcenter is the point where all three perpendicular bisectors of a triangle's sides intersect. Perpendicular bisectors pass through the midpoint of each side at a right angle. In an acute scalene triangle, the circumcenter lies inside the triangle. To find it:
  • Determine the midpoint of each side of the triangle.
  • Draw a line perpendicular to the triangle side, passing through the midpoint (these are the perpendicular bisectors).
  • The intersection point of these lines gives you the circumcenter.
The circumcenter is significant as it is the center of the circumcircle - a circle that passes through all three vertices of the triangle, emphasizing the equality of its radii extending to each vertex.

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Most popular questions from this chapter

Find a counterexample to show that the conjecture is false. The difference of two numbers is always less than the greater number.

Cut two pieces of string that are each 24 centimeters long. Construct an isosceles triangle out of one string and a scalene triangle out of the other. Measure and record the side lengths. Then classify each triangle by its angles.

In Exercises \(19-22\) , tell whether the orthocenter is inside, on, or outside the triangle. Then \(\square\) nd the coordinates of the orthocenter. (See Example 3 ) $$\mathrm{T}(-2,1), \mathrm{U}(2,1), \mathrm{V}(0,4)$$

In \(\Delta \mathrm{EFG}\) , the bisector of \(\angle \mathrm{F}\) intersects the bisector of \(\angle \mathrm{G}\) at point \(\mathrm{H}\) . Explain why \(\overline{\mathrm{FG}}\) must be longer than \(\overline{\mathrm{FH}}\) or \(\overline{\mathrm{HG}}\)

In Exercises \(19-22\) , tell whether the orthocenter is inside, on, or outside the triangle. Then \(\square\) nd the coordinates of the orthocenter. (See Example 3 ) $$\mathrm{A}(-4,0), \mathrm{B}(1,0), \mathrm{C}(-1,3)$$
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