Mathematics
Grade 10
15 min
Construct the circumcenter or incenter of a triangle (Tutorial Only)
Construct the circumcenter or incenter of a triangle (Tutorial Only)
What you'll learn
- Identify patterns in data sets (like tables or charts) and use them to predict what might happen next at least 8 out of 10 times.
- Explain, in your own words, why predictions based on data may not always be 100% accurate, giving at least two different reasons.
- Apply a given rule or pattern to extend a numerical or visual sequence and predict the next three elements correctly.
- Solve simple word problems that require making a prediction based on a given trend or rate, showing your work and arriving at the correct answer 4 out of 5 times.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define circumcenter, incenter, perpendicular bisector, and angle bisector.
Differentiate between the construction methods for the circumcenter and the incenter.
Accurately construct the perpendicular bisector of a line segment using a compass and straightedge.
Accurately construct the angle bisector of an angle using a compass and straightedge.
Construct the circumcenter of any triangle by finding the point of concurrency of the perpendicular bisectors.
Construct the incenter of any triangle by finding the point of concurrency of the angle bisectors.
Explain the key property of a circumcenter (equidistant from vertices) and an incenter (equidistant from sides).
Imagine a city wants to build a new library that is the exact same distance from three differ...
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Key Concepts & Vocabulary
TermDefinitionExample
Perpendicular BisectorA line, segment, or ray that is perpendicular to a segment at its midpoint.For a segment AB with midpoint M, the line passing through M such that it forms a 90° angle with AB is the perpendicular bisector.
Angle BisectorA ray that divides an angle into two adjacent, congruent angles.If ray BX bisects ∠ABC, then m∠ABX = m∠XBC.
Point of ConcurrencyThe point where three or more lines, rays, or segments intersect.The intersection point of the three perpendicular bisectors of a triangle is a point of concurrency.
CircumcenterThe point of concurrency of the three perpendicular bisectors of the sides of a triangle. It is the center of the triangle's circumscribed circle.In ΔABC, the point P where the perpendicular bisectors of sides AB, BC, and AC...
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Core Formulas
Circumcenter Theorem
If P is the circumcenter of ΔABC, then PA = PB = PC = r, where r is the radius of the circumscribed circle.
This theorem states that the circumcenter of a triangle is equidistant from the three vertices. This property is why it serves as the center of a circle that passes through all three vertices.
Incenter Theorem
If I is the incenter of ΔABC and ID ⊥ AB, IE ⊥ BC, IF ⊥ AC, then ID = IE = IF = r, where r is the radius of the inscribed circle.
This theorem states that the incenter of a triangle is equidistant from the three sides. This property is why it serves as the center of a circle that is tangent to all three sides.
Location of Centers
Circumcenter: Acute Δ (inside), Right Δ (on hypotenuse), Obtuse Δ (outside). Incenter: Always inside the tri...
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Challenging
A student correctly constructs the perpendicular bisectors of only two sides of a scalene triangle. Have they found enough information to locate the circumcenter?
A.Yes, because the third perpendicular bisector must intersect at the same point as the first two.
B.No, all three perpendicular bisectors must be constructed to ensure accuracy.
C.No, because the triangle is scalene, all three are needed.
D.Yes, but only if it is an acute triangle.
Challenging
In a special triangle, the circumcenter and the incenter are the exact same point. What kind of triangle must this be?
A.right triangle
B.An obtuse triangle
C.An equilateral triangle
D.An isosceles triangle that is not equilateral
Challenging
A right triangle has vertices at (0,0), (10,0), and (0,4). What are the coordinates of its circumcenter?
A.(0, 2)
B.(5, 2)
C.(5, 0)
D.(2.5, 2)
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Frequently asked questions
What grade level is "Construct the circumcenter or incenter of a triangle (Tutorial Only)"?
Construct the circumcenter or incenter of a triangle (Tutorial Only) is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Construct the circumcenter or incenter of a triangle (Tutorial Only)?
You'll be able to: Identify patterns in data sets (like tables or charts) and use them to predict what might happen next at least 8 out of 10 times; Explain, in your own words, why predictions based on data may not always be 100% accurate, giving….
Is "Construct the circumcenter or incenter of a triangle (Tutorial Only)" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Construct the circumcenter or incenter of a triangle (Tutorial Only)?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.