Mathematics
Grade 10
15 min
Midpoints
Midpoints
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1
Introduction & Learning Objectives
Learning Objectives
Define a midpoint and a segment bisector accurately.
Calculate the midpoint of a segment on a one-dimensional number line.
Derive and apply the Midpoint Formula to find the midpoint of a segment in the coordinate plane.
Determine the coordinates of an endpoint of a segment given the coordinates of its midpoint and the other endpoint.
Apply the Midpoint Formula to solve problems involving geometric figures like triangles and quadrilaterals.
Use the Midpoint Formula as a tool in coordinate geometry proofs.
Ever tried to hang a picture perfectly in the center of a wall or meet a friend exactly halfway? 🗺️ The concept of a midpoint is the mathematical key to finding that perfect central point.
In this tutorial, we will explore the concept of a midpoint, the...
2
Key Concepts & Vocabulary
TermDefinitionExample
Line SegmentA part of a line that is bounded by two distinct endpoints and contains every point on the line between its endpoints.The line segment AB, written as AB, consists of endpoints A and B and all points on the line between A and B.
MidpointThe point on a line segment that divides it into two congruent (equal length) segments.If M is the midpoint of segment AB, then the length of AM is equal to the length of MB (AM ≅ MB).
Congruent SegmentsLine segments that have the exact same length.If segment CD has a length of 5 cm and segment EF has a length of 5 cm, then CD and EF are congruent segments.
Segment BisectorAny point, line, ray, or other segment that intersects a line segment at its midpoint, thereby dividing it into two equal parts.If line L passes through...
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Core Formulas
Midpoint on a Number Line
M = \frac{x_1 + x_2}{2}
To find the coordinate of the midpoint of a segment on a number line, add the coordinates of the endpoints and divide by 2. This is simply finding the average of the two numbers.
Midpoint Formula (Coordinate Plane)
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
To find the coordinates of the midpoint of a segment in the coordinate plane, find the average of the x-coordinates and the average of the y-coordinates separately. The result is an ordered pair (x, y).
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Challenging
For a coordinate proof of the Triangle Midsegment Theorem, a triangle has vertices A(0,0), B(2b, 2c), and C(2a, 0). What are the coordinates of D, the midpoint of AB, and E, the midpoint of BC?
A.D(b, c) and E(a+b, c)
B.D(b, c) and E(a, 0)
C.D(2b, 2c) and E(2a+2b, 2c)
D.D(b, c) and E(a, c)
Challenging
The midpoints of the three sides of a triangle are P(2, 5), Q(6, 5), and R(4, 2). What are the coordinates of the vertices of the triangle?
A.(0,2), (4,8), (8,2)
B.(2,8), (6,2), (0,2)
C.(0, 2), (8, 8), (4, -4)
D.(1, 3), (5, 3), (3, 0)
Challenging
In a 3D coordinate system, the midpoint formula is an extension of the 2D formula. What is the midpoint of a segment with endpoints A(1, -3, 6) and B(7, 5, -2)?
A.(3, 4, 4)
B.(4, 1, 2)
C.(8, 2, 4)
D.(6, 8, -8)
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