Bisector Blueprint Worksheet
High School Geometry • Concurrency & Constructions
Bisector Blueprint Worksheet
Perpendicular Bisectors, Equidistance Theorems, & Triangle Circumcenters
Name
Date
Period
Score
Perpendicular Bisector
Perpendicular to a segment at its midpoint, dividing it into two equal halves at \(90^\circ\).
Equidistance Theorem
A point is on the \(\perp\) bisector of \(\overline{AB}\) iff it is equidistant from endpoints: \(PA = PB\).
The Circumcenter
Intersection of the 3 \(\perp\) bisectors. Equidistant from all 3 vertices (\(OA = OB = OC = R\)).
1
Distance Reasoning & Algebraic Proofs
1.1 Line \(k\) is the \(\perp\) bisector of \(\overline{MN}\). If \(PM = 5x - 7\) and \(PN = 2x + 14\), solve for \(x\) and find length \(PM\).
k P M N
Show algebraic setup & solution:
Value of \(x\): ________ Length \(PM\): ________
1.2 Point \(O\) is the circumcenter of \(\triangle ABC\). If \(OA = 4y - 3\) and \(OB = 2y + 11\), determine the circumradius \(R\) and find \(OC\).
O A B C
Show calculation & theorem justification:
Radius \(R\): ________ Length \(OC\): ________
2
Geometric Construction & Circumcenter Location
2.1 Compass Construction Task: Using a compass and straightedge, construct the perpendicular bisectors of sides \(\overline{AB}\) and \(\overline{BC}\) in acute \(\triangle ABC\) below.
- Leave all construction arcs clearly visible to verify proper technique.
- Label the intersection of the two perpendicular bisectors as circumcenter \(O\).
- Open your compass to \(OA\) and draw the circumscribed circle passing through all three vertices.
Construction Check: Does your circle pass cleanly through vertex \(C\)? If not, recheck your arc intersections.
A B C Construct \(\perp\) bisectors of \(\overline{AB}\) and \(\overline{BC}\)
2.2 Construction Justification: Why are only two perpendicular bisectors needed to locate the circumcenter of a triangle?
Geometry Mastery • Concurrency & Circumcenters Page 1 of 2
Bisector Blueprint Worksheet | Page 2: Analytical & Real-World Applications
Student: _______________________
3
Coordinate Geometry: Analytical Circumcenter Calculation
3.1 Step-by-Step Coordinate Method: A triangle in the coordinate plane has vertices \(D(0, 0)\), \(E(8, 0)\), and \(F(2, 6)\). Calculate the exact coordinates of circumcenter \(O(x, y)\).
Step A: Bisector of \(\overline{DE}\)
1. Midpoint of \(\overline{DE}\):
2. Slope \(m\) & \(\perp\) slope \(m_{\perp}\):
Equation 1: ________
Step B: Bisector of \(\overline{DF}\)
1. Midpoint of \(\overline{DF}\):
2. Slope \(m_{DF}\) & \(\perp\) slope \(m_{\perp}\):
Equation 2: ________
Step C: Concurrency Point
Solve system of equations for \(O(x, y)\):
Circumcenter \(O\): ( ____ , ____ )
Verification Check: Compute distance \(OD = \sqrt{(x-0)^2 + (y-0)^2}\): Circumradius \(R = \) ____________
4
Real-World Optimization: Regional Medevac Heliport
4.1 The Tri-City Medical Network: Three regional trauma hospitals in Fairview, Greenfield, and Oakridge form a triangular district. Emergency dispatch needs to position a trauma helicopter base so that flight distance to all three emergency rooms is precisely identical.
A. Triangle Center Selection Which triangle center must the heliport be built at? Explain why based on its equidistance property:
B. Flight Corridor Classification If the heliport lies directly on the boundary segment connecting Fairview & Greenfield, what kind of triangle is formed?
Network Geography Fairview Greenfield Oakridge H Equal flight corridors (\(R\))
5
Critical Reasoning & Error Analysis
5.1 Triangle Type & Concurrency Location
Complete the rule for the circumcenter’s location:
| Triangle Type | Circumcenter Location |
|---|
| Acute Triangle | Strictly ____________ the triangle |
| Right Triangle | Exact midpoint of the ____________ |
| Obtuse Triangle | Strictly ____________ the triangle |
5.2 Spot & Correct the Flaw
Elena's Claim: "Since the perpendicular bisector bisects the segment, it must always pass through the opposite vertex of the triangle."
Is Elena correct? Explain her error and state for which triangles her claim is true:
Geometry Mastery • Concurrency & Circumcenters Page 2 of 2
