Practice
Pythagorean Theorem and Special Right Triangles Practice
Fifty original questions on right-triangle anatomy, Pythagorean reasoning, special ratios, exact radicals, and composite figures.
- Answered
- 0 / 50
- Correct
- 0
- Incorrect
- 0
- Accuracy
- 0%
Question 1
Explanation
The hypotenuse is defined as the side opposite the right angle and is the longest side.
- Reasoning
Locate the right-angle marker, then look directly across the triangle.
Question 2
Explanation
The acute angles are complementary: \(x+8+2x+1=90\), so \(3x=81\) and \(x=27\).
- Reasoning
Use 90 degrees for the two acute angles.
Question 3
Explanation
The Pythagorean theorem reserves c for the side opposite the right angle.
- Reasoning
Identify the geometry before substituting values.
Question 4
Explanation
The converse test places the longest side in the c position, so c is 12.
- Reasoning
Sort the three lengths before squaring.
Question 5
Explanation
By the Pythagorean theorem, \(c=\sqrt{5^2+12^2}=\sqrt{169}=13\).
- Reasoning
Square both legs, add, and take the positive square root.
Question 6
Explanation
By the Pythagorean theorem, \(c=\sqrt{8^2+15^2}=\sqrt{289}=17\).
- Reasoning
Square both legs, add, and take the positive square root.
Question 7
Explanation
By the Pythagorean theorem, \(c=\sqrt{7^2+24^2}=\sqrt{625}=25\).
- Reasoning
Square both legs, add, and take the positive square root.
Question 8
Explanation
By the Pythagorean theorem, \(c=\sqrt{9^2+40^2}=\sqrt{1681}=41\).
- Reasoning
Square both legs, add, and take the positive square root.
Question 9
Explanation
By the Pythagorean theorem, \(c=\sqrt{20^2+21^2}=\sqrt{841}=29\).
- Reasoning
Square both legs, add, and take the positive square root.
Question 10
Explanation
By the Pythagorean theorem, \(c=\sqrt{12^2+35^2}=\sqrt{1369}=37\).
- Reasoning
Square both legs, add, and take the positive square root.
Question 11
Explanation
By the Pythagorean theorem, \(c=\sqrt{11^2+60^2}=\sqrt{3721}=61\).
- Reasoning
Square both legs, add, and take the positive square root.
Question 12
Explanation
By the Pythagorean theorem, \(c=\sqrt{28^2+45^2}=\sqrt{2809}=53\).
- Reasoning
Square both legs, add, and take the positive square root.
Question 13
Explanation
The missing leg satisfies \(b^2=10^2-6^2=64\), so \(b=8\).
- Reasoning
Subtract the known leg square from the hypotenuse square, then take the positive root.
Question 14
Explanation
The missing leg satisfies \(b^2=15^2-9^2=144\), so \(b=12\).
- Reasoning
Subtract the known leg square from the hypotenuse square, then take the positive root.
Question 15
Explanation
The missing leg satisfies \(b^2=20^2-12^2=256\), so \(b=16\).
- Reasoning
Subtract the known leg square from the hypotenuse square, then take the positive root.
Question 16
Explanation
The missing leg satisfies \(b^2=25^2-15^2=400\), so \(b=20\).
- Reasoning
Subtract the known leg square from the hypotenuse square, then take the positive root.
Question 17
Explanation
The missing leg satisfies \(b^2=26^2-10^2=576\), so \(b=24\).
- Reasoning
Subtract the known leg square from the hypotenuse square, then take the positive root.
Question 18
Explanation
The missing leg satisfies \(b^2=34^2-16^2=900\), so \(b=30\).
- Reasoning
Subtract the known leg square from the hypotenuse square, then take the positive root.
Question 19
Explanation
The missing leg satisfies \(b^2=82^2-18^2=6400\), so \(b=80\).
- Reasoning
Subtract the known leg square from the hypotenuse square, then take the positive root.
Question 20
Explanation
Using the longest side 15, compare \(9^2+12^2=225\) with \(15^2=225\). They are equal, so the converse proves the triangle is right.
- Reasoning
Put the longest length in the c position and compare exact squares.
Question 21
Explanation
Using the longest side 26, compare \(10^2+24^2=676\) with \(26^2=676\). They are equal, so the converse proves the triangle is right.
- Reasoning
Put the longest length in the c position and compare exact squares.
Question 22
Explanation
Using the longest side 11, compare \(7^2+8^2=113\) with \(11^2=121\). They are unequal, so the converse does not establish a right triangle.
- Reasoning
Put the longest length in the c position and compare exact squares.
Question 23
Explanation
Using the longest side 21, compare \(12^2+16^2=400\) with \(21^2=441\). They are unequal, so the converse does not establish a right triangle.
- Reasoning
Put the longest length in the c position and compare exact squares.
Question 24
Explanation
Using the longest side 29, compare \(20^2+21^2=841\) with \(29^2=841\). They are equal, so the converse proves the triangle is right.
- Reasoning
Put the longest length in the c position and compare exact squares.
Question 25
Explanation
Using the longest side 13, compare \(8^2+10^2=164\) with \(13^2=169\). They are unequal, so the converse does not establish a right triangle.
- Reasoning
Put the longest length in the c position and compare exact squares.
Question 26
Explanation
A 45-45-90 triangle has ratio \(1:1:\sqrt2\), so the hypotenuse is \(3\sqrt2\).
- Reasoning
Confirm the equal legs and multiply one leg by square root of 2.
Question 27
Explanation
A 45-45-90 triangle has ratio \(1:1:\sqrt2\), so the hypotenuse is \(5\sqrt2\).
- Reasoning
Confirm the equal legs and multiply one leg by square root of 2.
Question 28
Explanation
A 45-45-90 triangle has ratio \(1:1:\sqrt2\), so the hypotenuse is \(7\sqrt2\).
- Reasoning
Confirm the equal legs and multiply one leg by square root of 2.
Question 29
Explanation
A 45-45-90 triangle has ratio \(1:1:\sqrt2\), so the hypotenuse is \(9\sqrt2\).
- Reasoning
Confirm the equal legs and multiply one leg by square root of 2.
Question 30
Explanation
A 45-45-90 triangle has ratio \(1:1:\sqrt2\), so the hypotenuse is \(11\sqrt2\).
- Reasoning
Confirm the equal legs and multiply one leg by square root of 2.
Question 31
Explanation
A 45-45-90 triangle has ratio \(1:1:\sqrt2\), so the hypotenuse is \(13\sqrt2\).
- Reasoning
Confirm the equal legs and multiply one leg by square root of 2.
Question 32
Explanation
A 45-45-90 triangle has ratio \(1:1:\sqrt2\), so the hypotenuse is \(16\sqrt2\).
- Reasoning
Confirm the equal legs and multiply one leg by square root of 2.
Question 33
Explanation
A 45-45-90 triangle has ratio \(1:1:\sqrt2\), so the hypotenuse is \(20\sqrt2\).
- Reasoning
Confirm the equal legs and multiply one leg by square root of 2.
Question 34
Explanation
The ratio is \(x:x\sqrt3:2x\), giving longer leg \(2\sqrt3\) and hypotenuse \(4\).
- Reasoning
Match each side to its opposite angle before applying the 1:√3:2 ratio.
Question 35
Explanation
The ratio is \(x:x\sqrt3:2x\), giving longer leg \(4\sqrt3\) and hypotenuse \(8\).
- Reasoning
Match each side to its opposite angle before applying the 1:√3:2 ratio.
Question 36
Explanation
The ratio is \(x:x\sqrt3:2x\), giving longer leg \(5\sqrt3\) and hypotenuse \(10\).
- Reasoning
Match each side to its opposite angle before applying the 1:√3:2 ratio.
Question 37
Explanation
The ratio is \(x:x\sqrt3:2x\), giving longer leg \(7\sqrt3\) and hypotenuse \(14\).
- Reasoning
Match each side to its opposite angle before applying the 1:√3:2 ratio.
Question 38
Explanation
The ratio is \(x:x\sqrt3:2x\), giving longer leg \(8\sqrt3\) and hypotenuse \(16\).
- Reasoning
Match each side to its opposite angle before applying the 1:√3:2 ratio.
Question 39
Explanation
The ratio is \(x:x\sqrt3:2x\), giving longer leg \(10\sqrt3\) and hypotenuse \(20\).
- Reasoning
Match each side to its opposite angle before applying the 1:√3:2 ratio.
Question 40
Explanation
The ratio is \(x:x\sqrt3:2x\), giving longer leg \(12\sqrt3\) and hypotenuse \(24\).
- Reasoning
Match each side to its opposite angle before applying the 1:√3:2 ratio.
Question 41
Explanation
The ratio is \(x:x\sqrt3:2x\), giving longer leg \(15\sqrt3\) and hypotenuse \(30\).
- Reasoning
Match each side to its opposite angle before applying the 1:√3:2 ratio.
Question 42
Explanation
The shared altitude 13 and 84 are the legs, while 85 is opposite the right angle. Therefore \(13^2+84^2=85^2\).
- Reasoning
Solve or identify the shared side, then reset the hypotenuse position for the second triangle.
Question 43
Explanation
The shared altitude 15 and 20 are the legs, while 25 is opposite the right angle. Therefore \(15^2+20^2=25^2\).
- Reasoning
Solve or identify the shared side, then reset the hypotenuse position for the second triangle.
Question 44
Explanation
The shared altitude 25 and 60 are the legs, while 65 is opposite the right angle. Therefore \(25^2+60^2=65^2\).
- Reasoning
Solve or identify the shared side, then reset the hypotenuse position for the second triangle.
Question 45
Explanation
The shared altitude 17 and 144 are the legs, while 145 is opposite the right angle. Therefore \(17^2+144^2=145^2\).
- Reasoning
Solve or identify the shared side, then reset the hypotenuse position for the second triangle.
Question 46
Explanation
The shared altitude 20 and 21 are the legs, while 29 is opposite the right angle. Therefore \(20^2+21^2=29^2\).
- Reasoning
Solve or identify the shared side, then reset the hypotenuse position for the second triangle.
Question 47
Explanation
The hypotenuse must be alone on the c-squared side, so the two leg squares equal 10 squared.
- Reasoning
Locate the right-angle marker and place its opposite side as c.
Question 48
Explanation
The special ratio is \(1:1:\sqrt2\), so the hypotenuse is \(6\sqrt2\), not twice the leg.
- Reasoning
Do not mix the 45-45-90 ratio with the 30-60-90 hypotenuse rule.
Question 49
Explanation
The sides are \(6,6\sqrt3,12\). Their sum is \(18+6\sqrt3\).
- Reasoning
Write all three sides from the ratio before adding.
Question 50
Explanation
The hypotenuse is \(\sqrt{5^2+7^2}=\sqrt{74}\), which cannot be simplified further.
- Reasoning
Take the square root after adding the leg squares and preserve exact form.
Keyboard: use Tab to move, arrow keys to change answer choices, and Enter to check an answer.
Your practice summary
Use the results to decide what to review before your next attempt.
- Correct
- 0
- Incorrect
- 0
- Completed
- 50 / 50
Questions to review
No mistakes this time. Excellent work.
- Question 1Right-triangle anatomyEasy
- Question 2Complementary acute anglesEasy
- Question 3Pythagorean setupEasy
- Question 4Pythagorean converseEasy
- Question 5Missing hypotenuseEasy
- Question 6Missing hypotenuseEasy
- Question 7Missing hypotenuseEasy
- Question 8Missing hypotenuseEasy
- Question 9Missing hypotenuseEasy
- Question 10Missing hypotenuseEasy
- Question 11Missing hypotenuseEasy
- Question 12Missing hypotenuseEasy
- Question 13Missing legEasy
- Question 14Missing legEasy
- Question 15Missing legEasy
- Question 16Missing legMedium
- Question 17Missing legMedium
- Question 18Missing legMedium
- Question 19Missing legMedium
- Question 20Pythagorean converseMedium
- Question 21Pythagorean converseMedium
- Question 22Pythagorean converseMedium
- Question 23Pythagorean converseMedium
- Question 24Pythagorean converseMedium
- Question 25Pythagorean converseMedium
- Question 2645-45-90 trianglesMedium
- Question 2745-45-90 trianglesMedium
- Question 2845-45-90 trianglesMedium
- Question 2945-45-90 trianglesMedium
- Question 3045-45-90 trianglesMedium
- Question 3145-45-90 trianglesMedium
- Question 3245-45-90 trianglesMedium
- Question 3345-45-90 trianglesMedium
- Question 3430-60-90 trianglesMedium
- Question 3530-60-90 trianglesMedium
- Question 3630-60-90 trianglesMedium
- Question 3730-60-90 trianglesMedium
- Question 3830-60-90 trianglesMedium
- Question 3930-60-90 trianglesMedium
- Question 4030-60-90 trianglesMedium
- Question 4130-60-90 trianglesHard
- Question 42Composite right trianglesHard
- Question 43Composite right trianglesHard
- Question 44Composite right trianglesHard
- Question 45Composite right trianglesHard
- Question 46Composite right trianglesHard
- Question 47Error analysisHard
- Question 48Special-triangle error analysisHard
- Question 49Special-triangle perimeterHard
- Question 50Exact radicalHard