Practice
Rectangles, Squares, and Trapezoids Practice
Fifty original questions on diagonal properties, special square relationships, trapezoid midsegments, heights, and areas.
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Question 1
Explanation
The four-right-angle condition defines a rectangle.
- Reasoning
Combine diagonal congruence with bisection only when needed.
Question 2
Explanation
Rectangle diagonals have equal full lengths, and their intersection is the midpoint of both.
- Reasoning
Combine diagonal congruence with bisection only when needed.
Question 3
Explanation
The two diagonals divide the rectangle into four triangles of equal area.
- Reasoning
Combine diagonal congruence with bisection only when needed.
Question 4
Explanation
The diagonals bisect each other, so each vertex-to-intersection segment is half of 26.
- Reasoning
Combine diagonal congruence with bisection only when needed.
Question 5
Explanation
The diagonal is the hypotenuse: \(d=\sqrt{6^2+8^2}=10\).
- Reasoning
Use the rectangle dimensions as perpendicular legs.
Question 6
Explanation
The diagonal is the hypotenuse: \(d=\sqrt{9^2+12^2}=15\).
- Reasoning
Use the rectangle dimensions as perpendicular legs.
Question 7
Explanation
The diagonal is the hypotenuse: \(d=\sqrt{12^2+16^2}=20\).
- Reasoning
Use the rectangle dimensions as perpendicular legs.
Question 8
Explanation
The diagonal is the hypotenuse: \(d=\sqrt{7^2+24^2}=25\).
- Reasoning
Use the rectangle dimensions as perpendicular legs.
Question 9
Explanation
The diagonal is the hypotenuse: \(d=\sqrt{10^2+24^2}=26\).
- Reasoning
Use the rectangle dimensions as perpendicular legs.
Question 10
Explanation
The diagonal is the hypotenuse: \(d=\sqrt{20^2+21^2}=29\).
- Reasoning
Use the rectangle dimensions as perpendicular legs.
Question 11
Explanation
Each of the four triangles has equal area, so \(96/4=24\).
- Reasoning
Divide the rectangle's total area by four.
Question 12
Explanation
Each of the four triangles has equal area, so \(140/4=35\).
- Reasoning
Divide the rectangle's total area by four.
Question 13
Explanation
Each of the four triangles has equal area, so \(168/4=42\).
- Reasoning
Divide the rectangle's total area by four.
Question 14
Explanation
Each of the four triangles has equal area, so \(220/4=55\).
- Reasoning
Divide the rectangle's total area by four.
Question 15
Explanation
A square combines four equal sides with four right angles; its diagonal is the hypotenuse of a 45-45-90 triangle.
- Reasoning
Separate side, diagonal, and area relationships.
Question 16
Explanation
A square combines four equal sides with four right angles; its diagonal is the hypotenuse of a 45-45-90 triangle.
- Reasoning
Separate side, diagonal, and area relationships.
Question 17
Explanation
A square combines four equal sides with four right angles; its diagonal is the hypotenuse of a 45-45-90 triangle.
- Reasoning
Separate side, diagonal, and area relationships.
Question 18
Explanation
A square combines four equal sides with four right angles; its diagonal is the hypotenuse of a 45-45-90 triangle.
- Reasoning
Separate side, diagonal, and area relationships.
Question 19
Explanation
Since \(d=s\sqrt2\), the side is 3. Therefore \(A=s^2=9\).
- Reasoning
Divide the diagonal by square root of 2 before squaring the side.
Question 20
Explanation
Since \(d=s\sqrt2\), the side is 5. Therefore \(A=s^2=25\).
- Reasoning
Divide the diagonal by square root of 2 before squaring the side.
Question 21
Explanation
Since \(d=s\sqrt2\), the side is 7. Therefore \(A=s^2=49\).
- Reasoning
Divide the diagonal by square root of 2 before squaring the side.
Question 22
Explanation
Since \(d=s\sqrt2\), the side is 9. Therefore \(A=s^2=81\).
- Reasoning
Divide the diagonal by square root of 2 before squaring the side.
Question 23
Explanation
Since \(d=s\sqrt2\), the side is 12. Therefore \(A=s^2=144\).
- Reasoning
Divide the diagonal by square root of 2 before squaring the side.
Question 24
Explanation
Since \(d=s\sqrt2\), the side is 15. Therefore \(A=s^2=225\).
- Reasoning
Divide the diagonal by square root of 2 before squaring the side.
Question 25
Explanation
The source convention uses exactly one parallel pair; area height must be perpendicular to those base lines.
- Reasoning
Read the parallel and right-angle markings rather than relying on appearance.
Question 26
Explanation
The source convention uses exactly one parallel pair; area height must be perpendicular to those base lines.
- Reasoning
Read the parallel and right-angle markings rather than relying on appearance.
Question 27
Explanation
The source convention uses exactly one parallel pair; area height must be perpendicular to those base lines.
- Reasoning
Read the parallel and right-angle markings rather than relying on appearance.
Question 28
Explanation
The source convention uses exactly one parallel pair; area height must be perpendicular to those base lines.
- Reasoning
Read the parallel and right-angle markings rather than relying on appearance.
Question 29
Explanation
The source convention uses exactly one parallel pair; area height must be perpendicular to those base lines.
- Reasoning
Read the parallel and right-angle markings rather than relying on appearance.
Question 30
Explanation
The midsegment is the average: \((8+16)/2=12\).
- Reasoning
Add the bases, then divide by two.
Question 31
Explanation
The midsegment is the average: \((11+19)/2=15\).
- Reasoning
Add the bases, then divide by two.
Question 32
Explanation
The midsegment is the average: \((14+24)/2=19\).
- Reasoning
Add the bases, then divide by two.
Question 33
Explanation
The midsegment is the average: \((17+31)/2=24\).
- Reasoning
Add the bases, then divide by two.
Question 34
Explanation
The midsegment is the average: \((20+34)/2=27\).
- Reasoning
Add the bases, then divide by two.
Question 35
Explanation
The midsegment is the average: \((23+37)/2=30\).
- Reasoning
Add the bases, then divide by two.
Question 36
Explanation
Congruent legs define the isosceles case and guarantee congruent diagonals and matching angles along each base.
- Reasoning
Require the congruent-leg condition before applying isosceles properties.
Question 37
Explanation
Congruent legs define the isosceles case and guarantee congruent diagonals and matching angles along each base.
- Reasoning
Require the congruent-leg condition before applying isosceles properties.
Question 38
Explanation
Congruent legs define the isosceles case and guarantee congruent diagonals and matching angles along each base.
- Reasoning
Require the congruent-leg condition before applying isosceles properties.
Question 39
Explanation
Congruent legs define the isosceles case and guarantee congruent diagonals and matching angles along each base.
- Reasoning
Require the congruent-leg condition before applying isosceles properties.
Question 40
Explanation
Congruent legs define the isosceles case and guarantee congruent diagonals and matching angles along each base.
- Reasoning
Require the congruent-leg condition before applying isosceles properties.
Question 41
Explanation
Use \(A=\tfrac12h(b_1+b_2)=\tfrac12(7)(10+18)=98\).
- Reasoning
Average the bases and multiply by the perpendicular height.
Question 42
Explanation
Use \(A=\tfrac12h(b_1+b_2)=\tfrac12(9)(12+24)=162\).
- Reasoning
Average the bases and multiply by the perpendicular height.
Question 43
Explanation
Use \(A=\tfrac12h(b_1+b_2)=\tfrac12(8)(15+27)=168\).
- Reasoning
Average the bases and multiply by the perpendicular height.
Question 44
Explanation
Use \(A=\tfrac12h(b_1+b_2)=\tfrac12(11)(16+30)=253\).
- Reasoning
Average the bases and multiply by the perpendicular height.
Question 45
Explanation
Use \(A=\tfrac12h(b_1+b_2)=\tfrac12(12)(21+35)=336\).
- Reasoning
Average the bases and multiply by the perpendicular height.
Question 46
Explanation
Use \(A=\tfrac12h(b_1+b_2)=\tfrac12(15)(24+40)=480\).
- Reasoning
Average the bases and multiply by the perpendicular height.
Question 47
Explanation
Each offset is \((24-14)/2=5\). A 5-12-13 triangle gives height 12, so \(A=\tfrac12(12)(14+24)=228\).
- Reasoning
Split the base difference equally, find height, then use the trapezoid formula.
Question 48
Explanation
The rectangle area is \(18(24)=432\), and the four diagonal triangles have equal area, so each has area 108.
- Reasoning
Find total rectangle area and divide by four.
Question 49
Explanation
A trapezoid midsegment is the arithmetic mean of the parallel base lengths.
- Reasoning
Remember average means sum divided by count.
Question 50
Explanation
The square area formula squares the side, while the diagonal equals side times square root of 2.
- Reasoning
Identify whether the given length is side or diagonal.
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Questions to review
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- Question 1Rectangle propertiesEasy
- Question 2Rectangle propertiesEasy
- Question 3Rectangle propertiesEasy
- Question 4Rectangle propertiesEasy
- Question 5Rectangle diagonalEasy
- Question 6Rectangle diagonalEasy
- Question 7Rectangle diagonalEasy
- Question 8Rectangle diagonalEasy
- Question 9Rectangle diagonalEasy
- Question 10Rectangle diagonalEasy
- Question 11Equal-area diagonal trianglesEasy
- Question 12Equal-area diagonal trianglesEasy
- Question 13Equal-area diagonal trianglesEasy
- Question 14Equal-area diagonal trianglesEasy
- Question 15Square propertiesEasy
- Question 16Square propertiesMedium
- Question 17Square propertiesMedium
- Question 18Square propertiesMedium
- Question 19Square diagonal and areaMedium
- Question 20Square diagonal and areaMedium
- Question 21Square diagonal and areaMedium
- Question 22Square diagonal and areaMedium
- Question 23Square diagonal and areaMedium
- Question 24Square diagonal and areaMedium
- Question 25Trapezoid anatomyMedium
- Question 26Trapezoid anatomyMedium
- Question 27Trapezoid anatomyMedium
- Question 28Trapezoid anatomyMedium
- Question 29Trapezoid anatomyMedium
- Question 30Trapezoid midsegmentMedium
- Question 31Trapezoid midsegmentMedium
- Question 32Trapezoid midsegmentMedium
- Question 33Trapezoid midsegmentMedium
- Question 34Trapezoid midsegmentMedium
- Question 35Trapezoid midsegmentMedium
- Question 36Isosceles trapezoidsMedium
- Question 37Isosceles trapezoidsMedium
- Question 38Isosceles trapezoidsMedium
- Question 39Isosceles trapezoidsMedium
- Question 40Isosceles trapezoidsMedium
- Question 41Trapezoid areaHard
- Question 42Trapezoid areaHard
- Question 43Trapezoid areaHard
- Question 44Trapezoid areaHard
- Question 45Trapezoid areaHard
- Question 46Trapezoid areaHard
- Question 47Isosceles trapezoid synthesisHard
- Question 48Rectangle decompositionHard
- Question 49Midsegment error analysisHard
- Question 50Square error analysisHard