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📝 Pythagorean Theorem (a² + b² = c²) (26 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 26 questions available

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What is Pythagorean Theorem (a² + b² = c²)?

Definition:
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (called legs), expressed by the formula $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse, and this theorem is fundamental for calculating distances and checking right angles in geometry and real-world applications.

Working:
To use the theorem, identify the right triangle's legs and hypotenuse, then plug the known values into the equation and solve for the unknown side; if finding the hypotenuse, calculate $c = \sqrt{a^2 + b^2}$ , and if finding a leg, calculate $a = \sqrt{c^2 - b^2}$ , ensuring all units are consistent, and this relationship works only for right-angled triangles.

Example:
A ladder 13 feet long leans against a wall, with the foot of the ladder 5 feet from the wall; to find how high the ladder reaches, use $5^2 + h^2 = 13^2$ , so $25 + h^2 = 169$ , hence $h^2 = 144$ , and $h = 12$ feet, meaning the ladder reaches 12 feet up the wall.

Reason:
The Pythagorean theorem is essential in construction, navigation, physics, and everyday measurements, as it provides a reliable method to compute distances, design structures, and solve spatial problems, making it one of the most widely used mathematical principles.

Easy

Medium

Hard

📝 All Pythagorean Theorem (a² + b² = c²) MCQs

Q1. A right triangle has a hypotenuse of 1313 cm and one leg of 55 cm. What is the length of the missing leg?

A.8 cm

B.10 cm

C.12 cm ✅

D.14 cm

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: For a right triangle, the squared lengths of the two legs add to the squared hypotenuse. Thus $5^2+b^2=13^2$ , so $25+b^2=169$ , giving $b^2=144$ and $b=12$ cm. The answer is therefore 12 cm.

Q2. A right triangle has hypotenuse 1717 units and one leg 88 units. Which expression correctly represents the missing leg $x$ ?

x=17-8

x=\sqrt{17^2-8^2}

✅

x=\sqrt{17^2+8^2}

x=17^2-8^2

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The hypotenuse is the longest side, so its square must equal the sum of the squares of both legs. Rearranging gives $x^2=17^2-8^2$ , and therefore $x=\sqrt{17^2-8^2}$ .

Q3. A right triangular support has a 2525-m cable as its hypotenuse and a vertical height of 77 m. How far is the base from the point directly below the cable's upper end?

A.18 m

B.24 m ✅

C.26 m

D.32 m

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Let the horizontal distance be $x$ . Since the cable forms the hypotenuse, $x^2+7^2=25^2$ . Therefore $x^2=625-49=576$ , so $x=24$ m. The result is also geometrically reasonable because the missing leg must be shorter than 25 m.

Q4. A student calculates the missing leg of a right triangle with hypotenuse 1010 cm and another leg 66 cm as $10-6=4$ cm. What is the best evaluation of this reasoning?

A.It is correct because the hypotenuse is always subtracted from a leg.

B.It is correct because right triangles use ordinary subtraction.

C.It is incorrect because the side lengths must be related through their squares. ✅

D.It is incorrect because the missing leg must be longer than the hypotenuse.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The student's subtraction ignores the squared relationship among the side lengths. The correct setup is $x^2=10^2-6^2=64$ , giving $x=8$ cm. The missing leg is not found by simply subtracting the known leg from the hypotenuse.

Q5. A rectangular garden measures 1515 m by 2020 m. A diagonal divides the garden into two right triangles. If the diagonal is 2525 m, what is the shorter leg of either triangle?

A.10 m

B.15 m ✅

C.20 m

D.25 m

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The diagonal is the hypotenuse of each right triangle. Using $15^2+20^2=25^2$ , the given dimensions are already consistent. Therefore, the shorter leg is 15 m. The key modeling step is recognizing the rectangle's diagonal as the hypotenuse.

Q6. A wheelchair ramp forms a right triangle with a horizontal distance of 1212 ft and a ramp length of 1313 ft. If the vertical rise is $h$ , which value is most reasonable?

A.5 ft ✅

B.7 ft

C.12.5 ft

D.25 ft

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The ramp length is the hypotenuse, so $12^2+h^2=13^2$ . This gives $144+h^2=169$ , hence $h^2=25$ and $h=5$ ft. The result is reasonable because the rise must be shorter than the ramp.

Q7. A student uses $9^2+12^2=15^2$ to check a right triangle. Another student says the missing leg must be $15-12=3$ . Which conclusion is correct?

A.Both students are correct because both methods produce the same geometry.

B.The first student is correct; the missing leg is 9, not 3. ✅

C.The second student is correct because side lengths are always subtracted.

D.Neither student is correct because

9^2+12^2

is greater than

15^2

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The first calculation verifies that $9,12,15$ form a right triangle because $81+144=225$ . Subtracting 12 from 15 incorrectly gives 3, because the relationship involves squared lengths rather than direct subtraction.

Q8. A graph shows a right triangle with vertices at $(0,0)$ , $(6,0)$ , and $(0,8)$ . What is the length of the slanted side connecting $(6,0)$ and $(0,8)$ ?

A.7 units

B.8 units

C.10 units ✅

D.14 units

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The horizontal and vertical sides have lengths 6 and 8, forming the two legs of a right triangle. Thus the slanted side satisfies $c^2=6^2+8^2=36+64=100$ , so $c=10$ units.

Q9. A construction worker measures a diagonal brace as 2626 ft and the vertical side as 1010 ft. He needs the horizontal side before cutting material. What length should he use?

A.16 ft

B.24 ft ✅

C.28 ft

D.36 ft

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The diagonal brace is the hypotenuse, so if $x$ is the horizontal side, $x^2+10^2=26^2$ . Therefore $x^2=676-100=576$ , giving $x=24$ ft. This also satisfies the requirement that each leg be shorter than the hypotenuse.

Q10. A right triangle has a hypotenuse of 2020 units. One leg is 44 units longer than the other. What is the length of the shorter leg?

A.8 units ✅

B.10 units

C.12 units

D.16 units

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Let the shorter leg be $x$ , making the other $x+4$ . Then $x^2+(x+4)^2=20^2$ . Expanding gives $2x^2+8x-384=0$ , which simplifies to $x^2+4x-192=0$ . Factoring gives $(x+16)(x-12)=0$ , so the positive solution is 12 units.

Q11. Two right triangles have the same hypotenuse of 1313 units. Triangle A has one leg of 55 units, while Triangle B has one leg of 1212 units. Which comparison of their missing legs is correct?

A.Triangle A has missing leg 12, and Triangle B has missing leg 5. ✅

B.Both missing legs are 13.

C.Triangle A has missing leg 8, and Triangle B has missing leg 1.

D.The triangles cannot have the same hypotenuse.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For Triangle A, $x^2=13^2-5^2=144$ , so $x=12$ . For Triangle B, $x^2=13^2-12^2=25$ , so $x=5$ . The two triangles exchange their leg lengths while retaining the same hypotenuse.

Q12. A triangular frame has hypotenuse 2929 cm and one leg 2020 cm. A student claims the missing leg is 2121 cm after calculating $29^2-20^2=441$ . What mistake, if any, did the student make?

A.No mistake; 441 is already the missing side length.

B.The student should add

29^2

and

20^2

C.The student found the square of the missing leg but must take the square root. ✅

D.The student should subtract 29 from 20 instead.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The calculation $29^2-20^2=441$ correctly gives the square of the missing leg, not the leg itself. Taking the square root gives $x=\sqrt{441}=21$ cm. Recognizing the difference between a squared length and the actual length is essential.

Q13. A right triangle has hypotenuse 1010 units. Its two legs are positive integers, and one leg is exactly twice the other. What is the length of the shorter leg?

A.2 units

B.4 units ✅

C.6 units

D.8 units

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Let the shorter leg be $x$ , so the longer leg is $2x$ . Then $x^2+(2x)^2=10^2$ , giving $5x^2=100$ and $x^2=20$ . Thus $x=2\sqrt5$ , approximately 4.47, so none of the listed integer choices is correct. Therefore, under the stated integer condition, no solution exists.

Q14. A 13-ft ladder rests against a vertical wall. Its foot is 5 ft from the wall. Assuming the ground and wall meet at a right angle, how high up the wall does the ladder reach?

A.8 ft

B.10 ft

C.12 ft ✅

D.14 ft

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The ladder is the hypotenuse, so $5^2+h^2=13^2$ . Thus $h^2=169-25=144$ , giving $h=12$ ft. The 8-ft choice comes from subtracting lengths directly, while 10 and 14 do not satisfy the required right-triangle relationship.

Q15. A rectangular room is 9 m long and 12 m wide. A cable must run directly from one corner to the opposite corner. What is the minimum cable length needed?

A.15 m ✅

B.18 m

C.21 m

D.108 m

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The diagonal forms the hypotenuse of a right triangle with legs 9 m and 12 m. Therefore $d^2=9^2+12^2=225$ , so $d=15$ m. The larger alternatives confuse area, perimeter, or unnecessary extra distance with the direct diagonal.

Q16. A construction brace forms a right triangle with a horizontal base of 7.5 m and a vertical height of 10 m. If the brace must span directly between the two endpoints, approximately how long should it be?

A.12.5 m ✅

B.13.5 m

C.17.5 m

D.75 m

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The brace is the hypotenuse, so $c=\sqrt{7.5^2+10^2}=\sqrt{156.25}=12.5$ m. A common error is adding the legs to obtain 17.5 m, which gives the length of a two-segment path rather than the straight brace.

Q17. A worker moves 6 m east and then 8 m north to reach a control box. If the worker could instead travel straight from the starting point to the box, how much distance would be saved?

A.2 m

B.4 m ✅

C.10 m

D.14 m

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The original route is $6+8=14$ m. The direct route is the diagonal $d=\sqrt{6^2+8^2}=10$ m. Therefore, the saving is $14-10=4$ m. This requires comparing a path length with the shortest straight-line displacement.

Q18. Two ladders reach the same height on a wall. Ladder X has its base 6 ft from the wall, while Ladder Y has its base 8 ft from the wall. Which statement is necessarily true?

A.Ladder X is longer

B.Ladder Y is longer ✅

C.Both ladders have the same length

D.The comparison cannot be made without the wall height

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For the same vertical height, increasing the horizontal leg increases the hypotenuse. Thus Ladder Y must be longer because its base is farther from the wall. The wall height is not needed to determine which hypotenuse is greater.

Q19. A rectangular playground has side lengths 20 m and 21 m. A student claims its diagonal must be 41 m because a diagonal connects two sides. Which response best evaluates the claim?

A.Correct, because diagonals equal the sum of adjacent sides

B.Correct, because the diagonal is always the perimeter divided by two

C.Incorrect, because the diagonal is about 29 m ✅

D.Incorrect, because the diagonal is exactly 20 m

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The diagonal satisfies $d^2=20^2+21^2=841$ , so $d=29$ m. Adding the side lengths gives 41 m, which represents one possible two-side path, not the straight-line diagonal across the rectangle.

Q20. A triangular support has perpendicular sides of lengths $x$ and 15 m and a brace of length 17 m. Which equation correctly represents the situation?

x^2+17^2=15^2

x^2+15^2=17^2

✅

x+15=17^2

x^2+15=17

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The brace is opposite the right angle, making it the hypotenuse. Therefore the two perpendicular legs are $x$ and 15, giving $x^2+15^2=17^2$ . Correctly identifying the hypotenuse is essential before substituting values into the relationship.

Q21. A city map shows two roads meeting at a right angle. A delivery driver travels 9 km along one road and 40 km along the other. Why is the straight-line distance between the start and destination less than 49 km?

A.Because straight-line distance ignores both roads

B.Because the diagonal is the hypotenuse and is shorter than the sum of the legs ✅

C.Because 49 km is always the maximum possible distance

D.Because the roads have different units

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The two road segments are perpendicular legs of a right triangle. Their sum, 49 km, describes the two-part route, while the direct distance is the hypotenuse $c=\sqrt{9^2+40^2}=41$ km. Thus the diagonal is shorter.

Q22. A student calculates the missing leg of a right triangle with hypotenuse 25 cm and other leg 7 cm as $25-7=18$ cm. What is the most important correction?

A.Subtract the legs only after squaring them

B.Use

x=\sqrt{25^2-7^2}

, giving 24 cm ✅

C.Use

x=25^2-7^2

, giving 576 cm

D.Add 25 and 7 before taking the square root

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The missing leg must be found from $x^2+7^2=25^2$ , so $x=\sqrt{625-49}=\sqrt{576}=24$ cm. Direct subtraction ignores the squared relationship and therefore does not preserve the geometry of the right triangle.

Q23. A ladder is 10 ft long and its foot is 6 ft from a wall. A student says the ladder reaches 8 ft because $10-6=4$ and then doubles the result. What is wrong with this reasoning?

A.The ladder is not the hypotenuse

B.The horizontal distance should be squared before subtraction ✅

C.The vertical height should be added to the ladder

D.The wall cannot form a right triangle

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The correct setup is $6^2+h^2=10^2$ . Hence $h=\sqrt{100-36}=\sqrt{64}=8$ ft. Although the student obtained the correct numerical answer accidentally, the reasoning is invalid because subtracting and doubling does not follow the geometric relationship.

Q24. A coordinate grid shows a rectangular garden with opposite corners at $(2,3)$ and $(14,8)$ . Which expression gives the direct distance between these corners?

\sqrt{12^2+5^2}

✅

12+5

\sqrt{14^2+8^2}

\sqrt{12^2-5^2}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The horizontal change is $14-2=12$ and the vertical change is $8-3=5$ . These perpendicular changes form the legs of a right triangle, so the direct distance is $\sqrt{12^2+5^2}=13$ . The other expressions misuse coordinates or the required squared relationship.

Q25. A rectangular storage area measures 15 ft by 20 ft. A diagonal safety cable is installed, but 2 ft of extra cable is required at each end for fastening. What total cable length should be purchased at minimum?

A.23 ft

B.25 ft

C.29 ft

D.33 ft ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: First find the diagonal: $d=\sqrt{15^2+20^2}=\sqrt{625}=25$ ft. The fastening requires $2+2=4$ additional feet, so the total is $25+4=29$ ft. Therefore, option C is correct; the other choices result from ignoring or misapplying the extra cable.

Q26. A ladder is positioned so that its foot is moved 4 ft farther from a wall while the top remains 3 ft lower. If the original ladder length was 15 ft and both positions form right triangles, what was the original distance of the foot from the wall?

A.6 ft

B.9 ft ✅

C.12 ft

D.15 ft

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Let the original horizontal distance be $x$ and height be $h$ . Then $x^2+h^2=225$ . After moving the foot, $(x+4)^2+(h-3)^2=225$ . Subtracting gives $8x-6h+25=0$ , or $4x-3h=-12.5$ . Combining this with the original equation yields $x=9$ ft, after solving the resulting system.

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