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MathChapter 18: Polygons

Notes

Rectangles, Squares, and Trapezoids Notes

Connect diagonal, area, midsegment, and special-triangle properties across rectangles, squares, and source-defined trapezoids.

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These shapes share familiar appearances, but SAT solutions must come from stated definitions and markings. Start by naming the shape and its guaranteed properties.

Definitions and guaranteed structure

Rectangle, square, and trapezoid comparison
ShapeDefinitionDiagonal facts
RectangleQuadrilateral with four right anglesCongruent and bisect each other
SquareFour right angles and four congruent sidesCongruent and bisect each other
TrapezoidExactly one pair of parallel sidesNo general congruence conclusion
Isosceles trapezoidA trapezoid with congruent legsDiagonals are congruent

Rectangle diagonals

Rectangle diagonalsRectangle ABCD has four right corners and congruent diagonals AC and BD that bisect each other at E.
Rectangle diagonalsRectangle ABCD has four right corners and congruent diagonals AC and BD that bisect each other at E.ABCDEdiagonals are congruent and bisect each other
Four equal-area trianglesThe diagonals of rectangle ABCD intersect at E and divide it into four triangles labeled 1 through 4 with equal area.
Four equal-area trianglesThe diagonals of rectangle ABCD intersect at E and divide it into four triangles labeled 1 through 4 with equal area.1234ABCDEArea 1 = Area 2 = Area 3 = Area 4
Rectangle area and diagonal
\[A=\ell w,\qquad d^2=\ell^2+w^2\]

A diagonal is the hypotenuse of a right triangle whose legs are the rectangle dimensions.

Worked example

Half a rectangle diagonal

A rectangle is 12 by 16. Its diagonals meet at E. Find the distance from E to a vertex.

  1. Full diagonal

    \(d=\sqrt{12^2+16^2}=20\).

  2. Bisect

    E is the diagonal midpoint, so the requested distance is \(20/2\).

10 units.

Squares and 45-45-90 diagonals

Square propertiesSquare ABCD has four congruent sides, four right angles, and congruent diagonals that bisect each other at E.
Square propertiesSquare ABCD has four congruent sides, four right angles, and congruent diagonals that bisect each other at E.ABCDEdiagonals are congruent and bisect each other
Square area and diagonal
\[A=s^2,\qquad d=s\sqrt2\]

A square diagonal forms two 45-45-90 triangles.

Worked example

Area from a square diagonal

A square has diagonal \(14\sqrt2\). Find its area.

  1. Side

    \(s=d/\sqrt2=14\).

  2. Area

    \(A=14^2=196\).

196 square units.

Trapezoid anatomy and midsegment

Trapezoid
For this chapter, a quadrilateral with exactly one pair of parallel sides. The parallel sides are bases; the nonparallel sides are legs.
Trapezoid anatomyTrapezoid ABCD has exactly one pair of parallel sides labeled bases b1 and b2, two nonparallel legs, and perpendicular height h.
Trapezoid anatomyTrapezoid ABCD has exactly one pair of parallel sides labeled bases b1 and b2, two nonparallel legs, and perpendicular height h.b₂b₁hbases are parallel; legs are nonparallelABCD
Trapezoid midsegmentM and N are marked midpoints of the legs of trapezoid ABCD. Segment MN is parallel to both bases and equals their average length.
Trapezoid midsegmentM and N are marked midpoints of the legs of trapezoid ABCD. Segment MN is parallel to both bases and equals their average length.b₂b₁MNMN = (b₁ + b₂)/2ABCD
Trapezoid midsegment
\[m=\frac{b_1+b_2}{2}\]

The segment joining the leg midpoints is parallel to both bases and has their average length.

Worked example

Recover a missing base

A trapezoid has midsegment 17 and one base 12. Find the other base.

  1. Average

    \(17=(12+b_2)/2\).

  2. Solve

    \(34=12+b_2\), so \(b_2=22\).

The other base is 22.

Isosceles trapezoids

Isosceles trapezoidAn isosceles trapezoid has parallel bases, congruent legs, congruent diagonals, and two marked pairs of congruent base angles.
Isosceles trapezoidAn isosceles trapezoid has parallel bases, congruent legs, congruent diagonals, and two marked pairs of congruent base angles.b₂b₁legs and diagonals congruent; base-angle pairs congruentABCD

Trapezoid area

Trapezoid areaA trapezoid shows parallel bases b1 and b2 and perpendicular distance h between their lines.
Trapezoid areaA trapezoid shows parallel bases b1 and b2 and perpendicular distance h between their lines.b₂b₁hA = ½h(b₁ + b₂)ABCD
Trapezoid area
\[A=\frac12h(b_1+b_2)=mh\]

The second form follows because the midsegment is the average of the bases. Height is perpendicular distance between base lines.

Worked example

Height from a right triangle

An isosceles trapezoid has bases 14 and 24 and legs 13. Find its area.

  1. Horizontal offset

    Symmetry splits the base difference: \((24-14)/2=5\).

  2. Height

    The leg, offset, and height form a 5-12-13 triangle, so \(h=12\).

  3. Area

    \(A=\tfrac12(12)(14+24)=228\).

228 square units.
Mini check

Midsegment and area

Bases are 10 and 18 and height is 7. Find the midsegment and area.

Show answer and explanation

Answer: Midsegment 14; area 98.

Average the bases, then use \(A=mh\).

Key takeaways

Key takeaways

  • Rectangle diagonals are congruent and bisect each other; their intersection creates four equal-area triangles.
  • A square has \(A=s^2\) and diagonal \(s\sqrt2\).
  • This source uses exactly one parallel-side pair to define a trapezoid.
  • The trapezoid midsegment averages the bases, and area equals midsegment times perpendicular height.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.