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Pythagorean theoremGeometryMiddle schoolabout 1:31

A right triangle has hypotenuse 13 and one leg 5. Find the other leg: animated lesson

A narrated, step-by-step pythagorean theorem video lesson with captions. Final answer: b = 12; area = 30; perimeter ≈ 30; angles ≈ 22.619865°, 67.380135°, 90°.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.6 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
  2. 2. Steps, part 1
    1. a2+b2=c2  ⇒  b2=c2−a2a^2 + b^2 = c^2 \;\Rightarrow\; b^2 = c^2 - a^2
      Rearrange the Pythagorean theorem for a leg
    2. b2=132−52=169−25=144b^2 = 13^2 - 5^2 = 169 - 25 = 144
      Substitute and simplify
    3. b=144=12b = \sqrt{144} = 12
      Take the positive square root
    4. Area=12ab≈30,P=a+b+c≈30\text{Area} = \tfrac12ab \approx 30,\qquad P = a + b + c \approx 30
      Area and perimeter
  3. 3. Steps, part 2
    1. ∠A=arctan⁡ab≈22.619865∘,∠B=90∘−∠A≈67.380135∘\angle A = \arctan\frac{a}{b} \approx 22.619865^\circ,\qquad \angle B = 90^\circ - \angle A \approx 67.380135^\circ
      Acute angles
  4. 4. The answer
    b=12b = 12

    b = 12

  5. 5. Check
    • Check: a² + b² = 169 and c² = 169, which agree.
    • (5, 12, 13) is a Pythagorean triple: all three sides are whole numbers.
    • Angle sum: 90° + 22.619865° + 67.380135° = 180°.

Transcript

The problem. Here is the problem: A right triangle has hypotenuse 13 and one leg 5. Find the other leg. Find the missing leg of a right triangle from the other leg and the hypotenuse.

Steps, part 1. Rearrange the Pythagorean theorem for a leg. Substitute and simplify. Take the positive square root. Area and perimeter.

Steps, part 2. Acute angles. Angle A is opposite leg a and angle B is opposite leg b; the right angle C is 90°.

The answer. So the answer is b equals 12; area equals 30; perimeter approximately 30; angles approximately 22.619865°, 67.380135°, 90°. In every right triangle the square on the hypotenuse equals the sum of the squares on the two legs. Knowing any two sides fixes the third, and the angles follow from the tangent ratio.

Check. Check: a squared plus b squared equals 169 and c squared equals 169, which agree. (5, 12, 13) is a Pythagorean triple: all three sides are whole numbers. Angle sum: 90° plus 22.619865° plus 67.380135° equals 180°.

About pythagorean theorem

Find a missing side of a right triangle with a² + b² = c².

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