Trapezoid Survey A Answer Key Mission Alpha
OFFICIAL ANSWER KEY & AUDIT GUIDE
INTERNAL USE ONLY
REF: KEY-TRAP-ALPHA-FINAL
Vertex A (-6, -4)
Vertex B (10, -4)
Vertex C (7, 4)
Vertex D (-2, 4)
Step 2: Distance Analysis
NON-ISOSCELES CHECK
Leg AD Length:
\(\sqrt{80} \approx 8.94\)
\(\sqrt{4^2 + 8^2}\)
Leg BC Length:
\(\sqrt{73} \approx 8.54\)
\(\sqrt{(-3)^2 + 8^2}\)
Conclusion: NOT ISOSCELES (\(8.94 \neq 8.54\))
Step 3: Slope Analysis
AB 0
CD 0
AD 2
BC \(-\frac{8}{3}\)
Identical slopes (0) confirm AB and CD are the parallel bases.
Steps 4 & 5: Midpoint & Midsegment
Midpoint M (AD): (-4, 0)
Midpoint N (BC): (8.5, 0)
MN Verification
12.5 Units
\(\frac{16 + 9}{2} = 12.5\)
Step 6: Final Area Calculation
Certified
Calculation Path:
\(A = 0.5 \times (16 + 9) \times 8\)
Total Survey Area
100.0
Square Units
© 2026 Mathematics Bureau Field Analysis Key: Alpha-NonIso Coordinate Geometry Unit
Trapezoid Survey Visual Mission Trapezoid Territory
Step 1: Primary Boundary Mapping
SURVEYOR: ________________________
DATE: ___________________________
Site Coordinates
A(-6, -4) | B(10, -4) | C(7, 4) | D(-2, 4)
Utilize the coordinate ledger below to plot the four primary marker points. Connect the markers in alphabetical order to establish the land reserve perimeter.
1050-5-10
-10-50510
Status
Mapping in Progress
Target Shape
Quadrilateral
Region
Sector Alpha
REF: GEO-TRAP-SITE-P1-REFINED Official Field Survey Document Mathematics Bureau
Trapezoid Survey Distance Analysis Page Trapezoid Territory
Step 2: Measuring Boundary Distances
SURVEYOR: ________________________
DATE: ___________________________
Site Coordinates
A(-6, -4) | B(10, -4) | C(7, 4) | D(-2, 4)
Distance Analysis
Distance Formula
\( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)
Determine if the non-parallel legs (AD and BC) are equal in length. This verifies if the nature reserve is an Isosceles Trapezoid .
Length of Leg AD:
Length of Leg BC:
Isosceles Check
Does the reserve have congruent non-parallel legs?
Distance Calculation Ledger
REF: GEO-TRAP-SITE-P2-REFINED Official Field Survey Document Mathematics Bureau
Trapezoid Survey Slope Analysis Page Trapezoid Territory
Step 3: Proving Parallelism
SURVEYOR: ________________________
DATE: ___________________________
Site Coordinates
A(-6, -4) | B(10, -4) | C(7, 4) | D(-2, 4)
Slope Analysis
Slope Formula
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Calculate the slope of all four boundaries. Identify which pair of opposite sides are parallel (identical slopes).
Slope AB:
Slope CD:
Slope AD:
Slope BC:
Final Verification
Which sides are the parallel bases?
Calculation Ledger: Show All Work
REF: GEO-TRAP-SITE-P3-REFINED Official Field Survey Document Mathematics Bureau
Trapezoid Survey Midsegment Analysis Page Trapezoid Territory
Step 4: Midsegment Property Proof
SURVEYOR: ________________________
DATE: ___________________________
Site Coordinates
A(-6, -4) | B(10, -4) | C(7, 4) | D(-2, 4)
Midsegment
Midsegment Theorem
\( L = \frac{b_1 + b_2}{2} \)
The midsegment length is exactly equal to half the sum of the parallel bases (AB and CD). Calculate the length of MN and verify this property.
Calculated Length MN:
Property Match?
Does your result match the average of the bases?
Match Verified
Midsegment Calculation Ledger
REF: GEO-TRAP-SITE-P4-REFINED Official Field Survey Document Mathematics Bureau
Trapezoid Survey Midpoint Analysis Page Trapezoid Territory
Step 5: Central Navigation Plotting
SURVEYOR: ________________________
DATE: ___________________________
Site Coordinates
A(-6, -4) | B(10, -4) | C(7, 4) | D(-2, 4)
Midpoint Analysis
Midpoint Formula
\( M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \)
A central boardwalk path is planned to connect the midpoints of the non-parallel legs. Locate the coordinates for midpoints M (of leg AD) and N (of leg BC).
Midpoint M (AD):
( __ , __ )
Midpoint N (BC):
( __ , __ )
Visual Verification
Do your calculated midpoints lie exactly on the grid lines of your Part 1 survey?
Midpoint Calculation Ledger
REF: GEO-TRAP-SITE-P5-REFINED Official Field Survey Document Mathematics Bureau
Trapezoid Survey Final Area Page Trapezoid Territory
Step 6: Final Area Documentation
SURVEYOR: ________________________
DATE: ___________________________
Site Coordinates
A(-6, -4) | B(10, -4) | C(7, 4) | D(-2, 4)
Area Report
Area Formula
\( A = \frac{1}{2}(b_1 + b_2)h \)
Note: \( h \) is the vertical height.
Determine the final square unit area of the reserve. Use the vertical height identified from your Page 1 coordinate survey.
Base 1:
Base 2:
Height:
Final Official Result
TOTAL RESERVE AREA
Square Units
Area Calculation Ledger
REF: GEO-TRAP-SITE-P6-REFINED Conclusion of Field Mission Mathematics Bureau
Trapezoid Survey B Plotting Page Trapezoid Territory
Step 1: Primary Boundary Mapping (Mission B)
SURVEYOR: ________________________
DATE: ___________________________
Site B Mission Intel
A(2, 2) | B(18, 2) | C(14, 10) | D(6, 10)
Utilize the coordinate ledger below to plot the four primary marker points. Connect the markers in alphabetical order to establish the Site B perimeter.
20151050
05101520
Status
Plotting Active
Grid Scale
1 Unit = 10m
Sector
Site Bravo
REF: GEO-TRAP-SITE-B-P1 Site B Deployment Strategy Mathematics Bureau
Trapezoid Survey B Answer Key Mission Bravo
OFFICIAL ANSWER KEY & AUDIT GUIDE
INTERNAL USE ONLY
REF: KEY-TRAP-BRAVO-FINAL
Vertex A (2, 2)
Vertex B (18, 2)
Vertex C (14, 10)
Vertex D (6, 10)
Step 2: Distance Analysis
ISOSCELES CONFIRMED
Leg AD Length:
\(\sqrt{80} \approx 8.94\)
\(\sqrt{4^2 + 8^2}\)
Leg BC Length:
\(\sqrt{80} \approx 8.94\)
\(\sqrt{(-4)^2 + 8^2}\)
Result: Congruent Legs (Isosceles)
Step 3: Slope Analysis
AB 0
CD 0
AD 2
BC -2
Slopes 0 and 0 confirm AB and CD as the parallel bases.
Steps 4 & 5: Midpoint & Midsegment
Midpoint M (AD): (4, 6)
Midpoint N (BC): (16, 6)
MN Verification
12 Units
\(\frac{16 + 8}{2} = 12\)
Step 6: Final Area Calculation
Verified
Calculation Path:
\(A = 0.5 \times (16 + 8) \times 8\)
Total Survey Area
96.0
Square Units
© 2026 Mathematics Bureau Field Analysis Key: Bravo-Isosceles Coordinate Geometry Unit
Trapezoid Survey B Distance Analysis Page Trapezoid Territory
Step 2: Measuring Boundary Distances (Mission B)
SURVEYOR: ________________________
DATE: ___________________________
Site B Coordinates
A(2, 2) | B(18, 2) | C(14, 10) | D(6, 10)
Distance Analysis
Distance Formula
\( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)
Determine if the non-parallel legs (AD and BC) are equal in length. This verifies if the Site B reserve is an Isosceles Trapezoid .
Length of Leg AD:
Length of Leg BC:
Isosceles Check
Does Site B have congruent non-parallel legs?
Distance Calculation Ledger (Site B)
REF: GEO-TRAP-SITE-B-P2-REFINED Official Field Survey Document Mathematics Bureau
Trapezoid Survey B Slope Analysis Page Trapezoid Territory
Step 3: Proving Parallelism (Mission B)
SURVEYOR: ________________________
DATE: ___________________________
Site B Coordinates
A(2, 2) | B(18, 2) | C(14, 10) | D(6, 10)
Slope Analysis
Slope Formula
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Calculate the slope of all four boundaries. Identify which pair of opposite sides are parallel (identical slopes).
Slope AB:
Slope CD:
Slope AD:
Slope BC:
Final Verification
Which sides are the parallel bases?
Calculation Ledger: Show All Work
REF: GEO-TRAP-SITE-B-P3-REFINED Official Field Survey Document Mathematics Bureau
Trapezoid Survey B Midsegment Page Trapezoid Territory
Step 4: Midsegment Theorem (Mission B)
SURVEYOR: ________________________
DATE: ___________________________
Site B Coordinates
A(2, 2) | B(18, 2) | C(14, 10) | D(6, 10)
Midsegment Check
Midsegment Length
\( L = \frac{b_1 + b_2}{2} \)
The midsegment length is exactly equal to half the sum of the parallel bases (AB and CD). Calculate the length of MN and verify this property.
Calculated Length MN:
Property Verified?
Does your result match the theorem formula?
Match Verified
Midsegment Calculation Ledger (Site B)
REF: GEO-TRAP-SITE-B-P4-REFINED Official Field Survey Document Mathematics Bureau
Trapezoid Survey B Midpoint Analysis Page Trapezoid Territory
Step 5: Central Navigation (Mission B)
SURVEYOR: ________________________
DATE: ___________________________
Site B Coordinates
A(2, 2) | B(18, 2) | C(14, 10) | D(6, 10)
Midpoint Analysis
Midpoint Formula
\( M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \)
A central boardwalk path is planned to connect the midpoints of the non-parallel legs. Locate the coordinates for midpoints M (of leg AD) and N (of leg BC).
Midpoint M (AD):
( __ , __ )
Midpoint N (BC):
( __ , __ )
Visual Verification
Do these calculated coordinates sit exactly between the vertices on your Site B survey grid?
Midpoint Calculation Ledger (Site B)
REF: GEO-TRAP-SITE-B-P5-REFINED Official Field Survey Document Mathematics Bureau
Trapezoid Survey B Final Area Page Trapezoid Territory
Step 6: Final Area Documentation (Mission B)
SURVEYOR: ________________________
DATE: ___________________________
Site B Coordinates
A(2, 2) | B(18, 2) | C(14, 10) | D(6, 10)
Area Report
Area Formula
\( A = \frac{1}{2}(b_1 + b_2)h \)
Use vertical height (h).
Determine the final square unit area of the reserve. Use the vertical height identified from your Site B coordinate survey.
Base 1:
Base 2:
Height:
Final Official Result
TOTAL SITE AREA
Square Units
Area Calculation Ledger (Site B)
REF: GEO-TRAP-SITE-B-P6-REFINED Official Field Survey Document Mathematics Bureau
