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📝 Pythagorean theorem find hypotenuse (14 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 14 questions available

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What is Pythagorean theorem find hypotenuse?

Definition:
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs), and to find the hypotenuse, we use the formula $c = \sqrt{a^2 + b^2}$ , where $a$ and $b$ are the lengths of the legs, and $c$ is the hypotenuse.

Working:
To find the hypotenuse, first identify the legs of the right triangle (the two sides forming the right angle), then square each leg length, add the squares together, and take the square root of the sum, so the hypotenuse is $c = \sqrt{a^2 + b^2}$ , and this formula works for any right triangle, providing a direct way to find the longest side.

Example:
If a right triangle has legs of 6 cm and 8 cm, the hypotenuse is $c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm, meaning the side opposite the right angle is 10 cm long.

Reason:
Finding the hypotenuse using the Pythagorean theorem is critical in various fields such as construction (to check square corners), navigation (to find direct distances), physics (to calculate resultant vectors), and many everyday problems where distances need to be measured indirectly.

Easy

Medium

Hard

📝 All Pythagorean theorem find hypotenuse MCQs

Q1. A right triangle has legs measuring $9$ cm and $12$ cm. What is the length of its hypotenuse?

A.13 cm

B.14 cm

C.15 cm ✅

D.16 cm

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The hypotenuse is the longest side opposite the right angle. Using $c^2=9^2+12^2=81+144=225$ , we obtain $c=\sqrt{225}=15$ cm. Therefore, 15 cm is the correct length.

Q2. A right triangle has perpendicular sides of $5$ m and $12$ m. Which calculation correctly determines the hypotenuse?

5+12=17

12^2-5^2=119

\sqrt{5^2+12^2}=13

✅

\sqrt{12^2-5^2}=\sqrt{119}

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: For a right triangle, the two perpendicular sides are the legs, so their squares must be added before taking the square root. Thus $\sqrt{5^2+12^2}=\sqrt{169}=13$ m.

Q3. Two right triangles have leg pairs $(6,8)$ and $(9,12)$ . Without calculating both hypotenuses completely, which conclusion is correct?

A.The first has the longer hypotenuse

B.The second has the longer hypotenuse ✅

C.Both have the same hypotenuse

D.There is not enough information

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The second triangle is a scale enlargement of the first because $9/6=12/8=1.5$ . Since corresponding sides increase by a factor of $1.5$ , its hypotenuse also becomes $1.5$ times as long.

Q4. A student says that a triangle with legs $8$ and $15$ has hypotenuse $23$ because the hypotenuse should equal the sum of the legs. What is the best evaluation?

A.Correct because the longest side equals both legs added

B.Incorrect because the squares of the legs must be added before taking a square root ✅

C.Correct only when both legs are integers

D.Incorrect because the hypotenuse is always the difference of the legs

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Adding the legs gives a possible path length, not the straight-line hypotenuse. The correct calculation is $c=\sqrt{8^2+15^2}=\sqrt{289}=17$ . The student's method ignores the right-triangle relationship.

Q5. A rectangular garden measures $18$ m by $24$ m. A worker wants to walk directly from one corner to the opposite corner. What distance will the worker travel?

A.30 m ✅

B.32 m

C.36 m

D.42 m

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The diagonal creates a right triangle whose legs are $18$ m and $24$ m. Therefore $d=\sqrt{18^2+24^2}=\sqrt{324+576}=\sqrt{900}=30$ m.

Q6. A wheelchair ramp rises $3$ ft vertically while extending $12$ ft horizontally. Safety equipment must cover the full sloping length. Approximately how long should the covering be?

A.12.0 ft

B.12.4 ft ✅

C.13.0 ft

D.15.0 ft

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The ramp forms a right triangle with legs $3$ ft and $12$ ft. Its length is $\sqrt{3^2+12^2}=\sqrt{153}\approx12.37$ ft, so approximately $12.4$ ft of covering is needed.

Q7. A drone moves from point $A$ to point $B$ , traveling $40$ m east and $30$ m north. If it flies directly instead, what distance does it cover?

A.50 m ✅

B.60 m

C.70 m

D.80 m

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The eastward and northward movements are perpendicular, forming the legs of a right triangle. The direct distance is $\sqrt{40^2+30^2}=\sqrt{2500}=50$ m, which is shorter than traveling along both directions.

Q8. A rectangular screen has width $16$ inches and height $30$ inches. A diagonal support is installed from one corner to the opposite corner. Which value is closest to the support's length?

A.32 inches

B.34 inches ✅

C.36 inches

D.46 inches

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The diagonal is the hypotenuse of a right triangle. Calculate $d=\sqrt{16^2+30^2}=\sqrt{256+900}=\sqrt{1156}=34$ inches. The result is longer than either side but shorter than their sum.

Q9. A student calculates the hypotenuse of a right triangle with legs $7$ and $24$ as $\sqrt{24^2-7^2}=\sqrt{527}$ . What error did the student make?

A.The student squared the wrong side

B.The student subtracted the leg squares instead of adding them ✅

C.The student should add the legs before squaring

D.The student should divide the squares

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: When finding the hypotenuse, both leg squares contribute positively to the squared hypotenuse. The correct setup is $c^2=7^2+24^2=49+576=625$ , giving $c=25$ , not $\sqrt{527}$ .

Q10. A coordinate grid shows two points $P(2,3)$ and $Q(14,8)$ . The horizontal change is $12$ units and the vertical change is $5$ units. What is the straight-line distance $PQ$ ?

A.13 units ✅

B.17 units

C.19 units

D.24 units

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The coordinate differences form perpendicular horizontal and vertical components. Thus $PQ=\sqrt{12^2+5^2}=\sqrt{144+25}=\sqrt{169}=13$ units. The graph's horizontal and vertical changes provide the two legs.

Q11. A rectangular field has dimensions $21$ m and $28$ m. One person walks along two sides while another walks directly across the field. How much shorter is the direct route?

A.7 m

B.14 m ✅

C.21 m

D.49 m

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The direct route is $\sqrt{21^2+28^2}=\sqrt{441+784}=\sqrt{1225}=35$ m. Walking along two sides requires $21+28=49$ m, so the direct route saves $49-35=14$ m. Therefore, option B is correct.

Q12. A student compares two possible right-triangle paths. Path A has legs $10$ and $24$ , while Path B has legs $12$ and $20$ . Which path has the shorter hypotenuse?

A.Path A, because

34<32

B.Path B, because

\sqrt{544}<\sqrt{676}

✅

C.They have equal hypotenuses

D.Path A, because

10+24

is smaller than

12+20

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For Path A, the hypotenuse is $\sqrt{10^2+24^2}=\sqrt{676}=26$ . For Path B, it is $\sqrt{12^2+20^2}=\sqrt{544}\approx23.32$ . Therefore, Path B has the shorter hypotenuse.

Q13. A right triangle has one leg $x$ and another leg $x+7$ , while its hypotenuse is $17$ . Which positive value of $x$ satisfies the situation?

A.5

B.8 ✅

C.10

D.12

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Set up $x^2+(x+7)^2=17^2$ . Expanding gives $2x^2+14x+49=289$ , so $x^2+7x-120=0$ . Factoring gives $(x+15)(x-8)=0$ , and the positive solution is $x=8$ .

Q14. A square has side length $10$ cm. A designer claims its diagonal is $20$ cm because the diagonal connects two sides of length $10$ cm. What is the correct conclusion?

A.The diagonal is

10

B.The diagonal is

15

C.The diagonal is

10\sqrt{2}

cm ✅

D.The diagonal is

20

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The diagonal divides the square into a right triangle with both legs equal to $10$ cm. Therefore $d^2=10^2+10^2=200$ , so $d=\sqrt{200}=10\sqrt{2}\approx14.14$ cm, disproving the claim.

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