Unlock your inner mathematician with our Special Segments of a Triangle Worksheet Quiz! This geometry special segments quiz is designed to test your knowledge of triangle special segments worksheet topics like medians, altitudes, and angle bisectors. You'll put your understanding of points of concurrency in triangles to the test and learn to name the point of concurrency shown in each figure. Keep track of your progress, compare scores, and see how you stack up against fellow quiz-takers. Ready to sharpen your skills? Check out study examples or review detailed explanations before diving in. Take the quiz now and challenge yourself!
What is the segment from a triangle's vertex to the midpoint of the opposite side called?
Angle bisector
Perpendicular bisector
Altitude
Median
A median of a triangle is defined as the line segment joining a vertex to the midpoint of the opposite side. It divides that side into two equal parts and all three medians in a triangle intersect at the centroid. The centroid is the balancing point of the triangle. Learn more.
What is a line segment from a vertex perpendicular to the opposite side or its extension called?
Angle bisector
Altitude
Median
Perpendicular bisector
An altitude of a triangle is a perpendicular dropped from a vertex to the line containing the opposite side. It may fall inside or outside the triangle depending on the triangle's type. All three altitudes are concurrent at the orthocenter. Learn more.
Which segment divides an angle of a triangle into two congruent angles?
Perpendicular bisector
Median
Altitude
Angle bisector
An angle bisector is a line or segment that splits an angle into two equal measures. In a triangle, the three internal angle bisectors meet at the incenter. The incenter is equidistant from all sides of the triangle. Learn more.
What is a line that is perpendicular to a side of the triangle and passes through its midpoint called?
Perpendicular bisector
Altitude
Angle bisector
Median
A perpendicular bisector of a triangle's side is a line perpendicular to that side at its midpoint. All three perpendicular bisectors meet at the circumcenter, which is equidistant from the vertices. The circumcenter is the center of the circumscribed circle. Learn more.
The point where the three medians of a triangle intersect is called what?
Incenter
Circumcenter
Centroid
Orthocenter
The centroid is the common intersection of the medians of a triangle. It is located two-thirds of the distance from each vertex to the opposite midpoint. The centroid balances the triangle and is often called its center of mass. Learn more.
Which point is the intersection of the angle bisectors of a triangle?
Incenter
Circumcenter
Orthocenter
Centroid
The incenter is the intersection of the three internal angle bisectors of a triangle. It is equidistant from all three sides, making it the center of the inscribed circle. The incenter always lies inside the triangle. Learn more.
The intersection point of the altitudes of a triangle is known as what?
Centroid
Incenter
Circumcenter
Orthocenter
The orthocenter is the point where all three altitudes of a triangle meet. Its location varies: inside for acute triangles, at a vertex for right triangles, and outside for obtuse triangles. It is one of the four classic triangle centers. Learn more.
Which point is formed by the intersection of a triangle's perpendicular bisectors?
Orthocenter
Incenter
Centroid
Circumcenter
The perpendicular bisectors of a triangle's sides are concurrent at the circumcenter. This point is equidistant from all three vertices and is the center of the circumscribed circle. Its position can be inside, on, or outside the triangle. Learn more.
In an obtuse triangle, where is the orthocenter located?
Outside the triangle
On the triangle
Inside the triangle
At the midpoint of one side
For an obtuse triangle, at least one altitude falls outside the triangle, so the orthocenter (the intersection of altitudes) is also outside. In acute triangles it is inside, and in right triangles it lies at the right-angled vertex. Learn more.
The circumcenter of a triangle lies inside the triangle if and only if the triangle is what?
Obtuse
Acute
Right
Isosceles
The circumcenter is the point equidistant from all vertices. It is inside the triangle only when the triangle is acute, on the hypotenuse midpoint when the triangle is right, and outside when the triangle is obtuse. Learn more.
In a right triangle, the circumcenter coincides with which point?
The triangle's centroid
The midpoint of the hypotenuse
The orthocenter
The incenter
In a right triangle, the hypotenuse is a diameter of the circumcircle, so the circumcenter is located at its midpoint. This is a special property of right triangles. Learn more.
A triangle's incenter is equidistant from which parts of the triangle?
Vertices
Sides
Circumcircle
Midpoints of sides
The incenter is equidistant from all three sides of the triangle, making it the center of the inscribed circle. It lies within the triangle. Distances to vertices vary in general. Learn more.
The centroid divides each median in what ratio, measured from the vertex to the centroid and from the centroid to the midpoint?
2:1
1:1
3:1
1:2
The centroid divides each median so that the segment from the vertex to the centroid is twice as long as the segment from the centroid to the midpoint. This 2:1 ratio is consistent for all three medians. Learn more.
Which of the following segments always lies entirely inside a triangle?
Altitude
Perpendicular bisector
Median
Angle bisector
A median connects a vertex to the midpoint of the opposite side, and always lies within the triangle. Altitudes and perpendicular bisectors may fall outside in obtuse cases, and angle bisectors always lie inside but can be considered on edges in degenerate cases. Learn more.
On the Euler line of a non-equilateral triangle, what is the ratio of distances from the centroid to the circumcenter and from the centroid to the orthocenter?
3:2
1:1
1:2
2:1
Euler's line connects the orthocenter, centroid, and circumcenter, with the centroid dividing the segment from orthocenter to circumcenter in a 2:1 ratio (orthocenter to centroid is twice centroid to circumcenter). Thus, the distance from centroid to circumcenter is half the distance from centroid to orthocenter. Learn more.
Which formula relates the sum of the squares of the medians of a triangle to the sum of the squares of its sides?
m_a² + m_b² + m_c² = 1/2(a² + b² + c²)
m_a² + m_b² + m_c² = a² + b² + c²
m_a² + m_b² + m_c² = 2(a² + b² + c²)
m_a² + m_b² + m_c² = (3/4)(a² + b² + c²)
Apollonius' theorem states that the sum of the squares of the three medians equals three?quarters of the sum of the squares of the sides. This elegant relation links medians directly to the sides. It is a key result in triangle geometry. Learn more.
What is the length of the angle bisector from vertex A in terms of sides b, c, and angle A?
(b c)/(b + c) · ?[2b² + 2c² ? a²]
½(b + c)
?[b c ((b + c)² ? a²)]/(b + c)
(2 b c cos(A/2))/(b + c)
The internal angle bisector length from vertex A is given by 2 b c cos(A/2) divided by (b + c). This formula derives from the Angle Bisector Theorem and the Law of Cosines. It is widely used in advanced triangle calculations. Learn more.
Which formula gives the radius r of the inscribed circle (inradius) of a triangle with area ? and semiperimeter s?
r = (a + b + c)/(2?)
r = 2?/s
r = ?/(a + b + c)
r = ?/s
The inradius of a triangle is given by the area ? divided by the semiperimeter s. This formula follows from equating ? = r · s to the standard area. It provides a direct link between side lengths and the inscribed circle. Learn more.
The length of the altitude from vertex A onto side BC can be expressed as which of these?
b sin C
2?/a
(a + b + c)/2
c sin B
The altitude from A to BC has length 2 times the triangle's area ? divided by side a (the base BC). This comes from ? = (1/2)a·h_a. Rearranging gives h_a = 2?/a. Learn more.
The barycentric coordinates of the incenter of a triangle ABC are proportional to which of the following triples?
(a, b, c)
(1, 1, 1)
(a², b², c²)
(sin A, sin B, sin C)
In barycentric coordinates, the incenter is represented by the weights proportional to the side lengths a, b, and c. This reflects the property that the incenter is closer to longer sides. Learn more.
The distance from the circumcenter O to side BC equals which expression in terms of circumradius R and angle A?
R cos A
R tan A
2R cos A
R sin A
Dropping a perpendicular from O to BC gives a distance equal to R cos A because OB = R and the angle between OB and the perpendicular to BC is A. This follows from basic right?triangle trigonometry. Learn more.
The reflection of the orthocenter H across the midpoint of side BC lies on which circle?
Incircle
Nine-point circle
Jerabek hyperbola
Circumcircle
When the orthocenter is reflected across the midpoint of a side, the reflection lies on the circumcircle at the antipode of the opposite vertex. This is a classical property of triangle centers. Learn more.
For triangle A(0,0), B(6,0), C(0,8), find the distance between its orthocenter and circumcenter.
4
5
10
?41
This is a right triangle at A, so the orthocenter coincides with A(0,0). The circumcenter is at the midpoint of BC, which is (3,4). The distance from (0,0) to (3,4) is ?(3²+4²)=5. Learn more.
In a triangle with side lengths a=13, b=14, c=15, what is the length of the median to side c?
?605/2
?455/2
?565/2
?505/2
The median to side c has length m_c = ½?(2a² + 2b² ? c²) = ½?(2·169 + 2·196 ? 225) = ½?505. This uses Apollonius' theorem. Learn more.
For a triangle with sides 13, 14, and 15, what is the radius of its inscribed circle (inradius)?
5
3.5
4
6
First compute semiperimeter s=(13+14+15)/2=21. The area by Heron's formula is ?=?(21·8·7·6)=84. Then r=?/s=84/21=4. This is the inradius. Learn more.
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Study Outcomes
Identify Special Segments -
Recognize medians, altitudes, angle bisectors, and perpendicular bisectors in triangle diagrams from the special segments of a triangle worksheet.
Define Key Properties -
Articulate characteristics and construction rules of each triangle special segment during the geometry special segments quiz.
Analyze Points of Concurrency -
Name the point of concurrency shown in various triangle special segments worksheet diagrams and explain its geometric significance.
Differentiate Concurrency Points -
Distinguish between centroid, circumcenter, incenter, and orthocenter in different triangle configurations.
Apply Construction Techniques -
Accurately draw and construct special segments of a triangle using straightedge and compass methods.
Evaluate Segment Relationships -
Assess relationships between special segments and points of concurrency in triangles to deepen geometry insights.
Cheat Sheet
Medians and Centroid -
Each median connects a vertex to the midpoint of the opposite side, and the three medians intersect at the centroid. The centroid divides each median in a 2:1 ratio, with the longer segment between the vertex and the centroid (MIT OpenCourseWare). Think of the centroid as the triangle's "center of mass," perfect for balancing or center-of-gravity problems.
Altitudes and Orthocenter -
An altitude is a perpendicular segment from a vertex to its opposite side or its extension, and all three altitudes converge at the orthocenter. In acute triangles the orthocenter lies inside, while in obtuse triangles it falls outside the shape (Khan Academy). A handy mnemonic: the orthocenter is the "height center," gathering all the triangle's heights.
Angle Bisectors and Incenter -
Angle bisectors split each vertex angle into two equal parts, meeting at the incenter, which is equidistant from all three sides and is the center of the inscribed circle. You can find its coordinates via (ax+bx₂+cx₃)/(a+b+c), weighting each vertex by the length of the opposite side (University of Cambridge). Remember "in" center goes inside and inscribes the largest possible circle.
Perpendicular Bisectors and Circumcenter -
Perpendicular bisectors of each side meet at the circumcenter, the point equidistant from all three vertices and the center of the circumscribed circle. Depending on triangle type, the circumcenter can lie inside (acute), on (right), or outside (obtuse) the triangle (Wolfram MathWorld). Mnemonic: "circum" center "circles" all the vertices.
Euler Line -
In any non-equilateral triangle, the centroid, orthocenter, and circumcenter are collinear on the Euler line. The centroid divides the segment between the orthocenter and circumcenter in a 2:1 ratio, closer to the circumcenter (University of Oxford). Recall "Euler" to remember this elegant alignment of key centers.
