Practice
Area of a Triangle Practice
Fifty original questions on base-height selection, equilateral area, area ratios, percentage changes, and Pythagorean connections.
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Question 1
Explanation
Using \(A=\tfrac12bh\), the area is \(\tfrac12(8)(7)=28\) square units.
- Reasoning
Verify the given height is perpendicular, then multiply base and height and divide by 2.
Question 2
Explanation
Using \(A=\tfrac12bh\), the area is \(\tfrac12(12)(9)=54\) square units.
- Reasoning
Verify the given height is perpendicular, then multiply base and height and divide by 2.
Question 3
Explanation
Using \(A=\tfrac12bh\), the area is \(\tfrac12(15)(6)=45\) square units.
- Reasoning
Verify the given height is perpendicular, then multiply base and height and divide by 2.
Question 4
Explanation
Using \(A=\tfrac12bh\), the area is \(\tfrac12(18)(11)=99\) square units.
- Reasoning
Verify the given height is perpendicular, then multiply base and height and divide by 2.
Question 5
Explanation
Using \(A=\tfrac12bh\), the area is \(\tfrac12(20)(13)=130\) square units.
- Reasoning
Verify the given height is perpendicular, then multiply base and height and divide by 2.
Question 6
Explanation
Using \(A=\tfrac12bh\), the area is \(\tfrac12(24)(5)=60\) square units.
- Reasoning
Verify the given height is perpendicular, then multiply base and height and divide by 2.
Question 7
Explanation
Using \(A=\tfrac12bh\), the area is \(\tfrac12(16)(14)=112\) square units.
- Reasoning
Verify the given height is perpendicular, then multiply base and height and divide by 2.
Question 8
Explanation
Using \(A=\tfrac12bh\), the area is \(\tfrac12(25)(12)=150\) square units.
- Reasoning
Verify the given height is perpendicular, then multiply base and height and divide by 2.
Question 9
Explanation
From \(A=\tfrac12bh\), the missing height is \(2A/14=2(84)/14=12\).
- Reasoning
Multiply the area by 2 before dividing by the known perpendicular dimension.
Question 10
Explanation
From \(A=\tfrac12bh\), the missing height is \(2A/18=2(108)/18=12\).
- Reasoning
Multiply the area by 2 before dividing by the known perpendicular dimension.
Question 11
Explanation
From \(A=\tfrac12bh\), the missing height is \(2A/10=2(75)/10=15\).
- Reasoning
Multiply the area by 2 before dividing by the known perpendicular dimension.
Question 12
Explanation
From \(A=\tfrac12bh\), the missing base is \(2A/12=2(96)/12=16\).
- Reasoning
Multiply the area by 2 before dividing by the known perpendicular dimension.
Question 13
Explanation
From \(A=\tfrac12bh\), the missing base is \(2A/15=2(135)/15=18\).
- Reasoning
Multiply the area by 2 before dividing by the known perpendicular dimension.
Question 14
Explanation
From \(A=\tfrac12bh\), the missing base is \(2A/14=2(126)/14=18\).
- Reasoning
Multiply the area by 2 before dividing by the known perpendicular dimension.
Question 15
Explanation
The equilateral formula gives \(A=(\sqrt3/4)(2)^2=1\sqrt3\).
- Reasoning
Square the side before multiplying by square root of 3 over 4.
Question 16
Explanation
The equilateral formula gives \(A=(\sqrt3/4)(4)^2=4\sqrt3\).
- Reasoning
Square the side before multiplying by square root of 3 over 4.
Question 17
Explanation
The equilateral formula gives \(A=(\sqrt3/4)(6)^2=9\sqrt3\).
- Reasoning
Square the side before multiplying by square root of 3 over 4.
Question 18
Explanation
The equilateral formula gives \(A=(\sqrt3/4)(8)^2=16\sqrt3\).
- Reasoning
Square the side before multiplying by square root of 3 over 4.
Question 19
Explanation
The equilateral formula gives \(A=(\sqrt3/4)(10)^2=25\sqrt3\).
- Reasoning
Square the side before multiplying by square root of 3 over 4.
Question 20
Explanation
The equilateral formula gives \(A=(\sqrt3/4)(12)^2=36\sqrt3\).
- Reasoning
Square the side before multiplying by square root of 3 over 4.
Question 21
Explanation
Areas scale with the square of the side factor, so square both terms: \(2^2:3^2=4:9\).
- Reasoning
Keep the order and square each part of the side ratio.
Question 22
Explanation
Areas scale with the square of the side factor, so square both terms: \(3^2:5^2=9:25\).
- Reasoning
Keep the order and square each part of the side ratio.
Question 23
Explanation
Areas scale with the square of the side factor, so square both terms: \(4^2:7^2=16:49\).
- Reasoning
Keep the order and square each part of the side ratio.
Question 24
Explanation
Areas scale with the square of the side factor, so square both terms: \(5^2:8^2=25:64\).
- Reasoning
Keep the order and square each part of the side ratio.
Question 25
Explanation
Areas scale with the square of the side factor, so square both terms: \(3^2:2^2=9:4\).
- Reasoning
Keep the order and square each part of the side ratio.
Question 26
Explanation
Areas scale with the square of the side factor, so square both terms: \(7^2:4^2=49:16\).
- Reasoning
Keep the order and square each part of the side ratio.
Question 27
Explanation
Areas scale with the square of the side factor, so square both terms: \(6^2:11^2=36:121\).
- Reasoning
Keep the order and square each part of the side ratio.
Question 28
Explanation
Areas scale with the square of the side factor, so square both terms: \(9^2:5^2=81:25\).
- Reasoning
Keep the order and square each part of the side ratio.
Question 29
Explanation
Areas scale with the square of the side factor, so square both terms: \(4^2:9^2=16:81\).
- Reasoning
Keep the order and square each part of the side ratio.
Question 30
Explanation
Areas scale with the square of the side factor, so square both terms: \(8^2:3^2=64:9\).
- Reasoning
Keep the order and square each part of the side ratio.
Question 31
Explanation
The common height and factor one-half cancel, leaving the base ratio \(3:8\).
- Reasoning
Equal-height area ratios are linear base ratios, not squared ratios.
Question 32
Explanation
The common height and factor one-half cancel, leaving the base ratio \(5:12\).
- Reasoning
Equal-height area ratios are linear base ratios, not squared ratios.
Question 33
Explanation
The common height and factor one-half cancel, leaving the base ratio \(7:10\).
- Reasoning
Equal-height area ratios are linear base ratios, not squared ratios.
Question 34
Explanation
The common height and factor one-half cancel, leaving the base ratio \(9:14\).
- Reasoning
Equal-height area ratios are linear base ratios, not squared ratios.
Question 35
Explanation
The common base and factor one-half cancel, leaving height ratio \(4:9\).
- Reasoning
Equal-base area ratios follow the heights directly.
Question 36
Explanation
The common base and factor one-half cancel, leaving height ratio \(6:11\).
- Reasoning
Equal-base area ratios follow the heights directly.
Question 37
Explanation
The common base and factor one-half cancel, leaving height ratio \(7:13\).
- Reasoning
Equal-base area ratios follow the heights directly.
Question 38
Explanation
The common base and factor one-half cancel, leaving height ratio \(10:17\).
- Reasoning
Equal-base area ratios follow the heights directly.
Question 39
Explanation
Area factors multiply: \(1.2)(0.75)=0.90\). Therefore the area has 10% decrease.
- Reasoning
Convert each percent change to a multiplier and multiply; do not add percentages.
Question 40
Explanation
Area factors multiply: \(1.5)(0.8)=1.20\). Therefore the area has 20% increase.
- Reasoning
Convert each percent change to a multiplier and multiply; do not add percentages.
Question 41
Explanation
Area factors multiply: \(1.1)(1.1)=1.21\). Therefore the area has 21% increase.
- Reasoning
Convert each percent change to a multiplier and multiply; do not add percentages.
Question 42
Explanation
Area factors multiply: \(1.25)(0.8)=1.00\). Therefore the area has no change.
- Reasoning
Convert each percent change to a multiplier and multiply; do not add percentages.
Question 43
Explanation
Area factors multiply: \(0.9)(1.3)=1.17\). Therefore the area has 17% increase.
- Reasoning
Convert each percent change to a multiplier and multiply; do not add percentages.
Question 44
Explanation
Area factors multiply: \(0.8)(0.75)=0.60\). Therefore the area has 40% decrease.
- Reasoning
Convert each percent change to a multiplier and multiply; do not add percentages.
Question 45
Explanation
Square areas equal side-length squares, so the Pythagorean theorem gives \(c^2=64+225=289\).
- Reasoning
Add the two leg-square areas directly.
Question 46
Explanation
The missing leg-square area is \(400-144=256\).
- Reasoning
Subtract a known leg-square area from the hypotenuse-square area.
Question 47
Explanation
Use \(30=\tfrac12(5)h\), so \(h=12\). The right-angle condition confirms the legs are a valid base-height pair.
- Reasoning
Solve the area equation before using any side-length theorem.
Question 48
Explanation
The hypotenuse is \(\sqrt{9^2+12^2}=15\), so perimeter is \(9+12+15=36\).
- Reasoning
Find the missing hypotenuse before adding all three sides.
Question 49
Explanation
A base and height form a 90-degree angle. A slanted side without perpendicular evidence is not the corresponding altitude.
- Reasoning
Find the right-angle marker before choosing b and h.
Question 50
Explanation
Area depends on two linear dimensions, so its scale factor is \(3^2=9\).
- Reasoning
Square the corresponding-side scale factor for similar-triangle areas.
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Questions to review
No mistakes this time. Excellent work.
- Question 1Triangle areaEasy
- Question 2Triangle areaEasy
- Question 3Triangle areaEasy
- Question 4Triangle areaEasy
- Question 5Triangle areaEasy
- Question 6Triangle areaEasy
- Question 7Triangle areaEasy
- Question 8Triangle areaEasy
- Question 9Missing base or heightEasy
- Question 10Missing base or heightEasy
- Question 11Missing base or heightEasy
- Question 12Missing base or heightEasy
- Question 13Missing base or heightEasy
- Question 14Missing base or heightEasy
- Question 15Equilateral triangle areaEasy
- Question 16Equilateral triangle areaMedium
- Question 17Equilateral triangle areaMedium
- Question 18Equilateral triangle areaMedium
- Question 19Equilateral triangle areaMedium
- Question 20Equilateral triangle areaMedium
- Question 21Similar-triangle area ratiosMedium
- Question 22Similar-triangle area ratiosMedium
- Question 23Similar-triangle area ratiosMedium
- Question 24Similar-triangle area ratiosMedium
- Question 25Similar-triangle area ratiosMedium
- Question 26Similar-triangle area ratiosMedium
- Question 27Similar-triangle area ratiosMedium
- Question 28Similar-triangle area ratiosMedium
- Question 29Similar-triangle area ratiosMedium
- Question 30Similar-triangle area ratiosMedium
- Question 31Same-height area ratioMedium
- Question 32Same-height area ratioMedium
- Question 33Same-height area ratioMedium
- Question 34Same-height area ratioMedium
- Question 35Same-base area ratioMedium
- Question 36Same-base area ratioMedium
- Question 37Same-base area ratioMedium
- Question 38Same-base area ratioMedium
- Question 39Percent change in areaMedium
- Question 40Percent change in areaMedium
- Question 41Percent change in areaHard
- Question 42Percent change in areaHard
- Question 43Percent change in areaHard
- Question 44Percent change in areaHard
- Question 45Pythagorean square areasHard
- Question 46Pythagorean square areasHard
- Question 47Right-triangle areaHard
- Question 48Area and perimeter with PythagoreanHard
- Question 49Base-height error analysisHard
- Question 50Area-scale error analysisHard