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Zorluk: OrtaPythagorean Theorem and Special Right Triangles

In quadrilateral ABCDABCD, ∠B=90∘\angle B = 90^\circ and ∠D=90∘\angle D = 90^\circ. If AB=12AB = 12 units, BC=16BC = 16 units, and AD=10AD = 10 units, what is the length, in units, of side CDCD?

  1. 10310\sqrt{3}Cevap
  2. B
    10510\sqrt{5}
  3. C
    6196\sqrt{19}
  4. D
    1818
  5. E
    1010

Cevap

10310\sqrt{3} units
Connecting vertex AA to vertex CC creates two right-angled triangles sharing hypotenuse ACAC. In right triangle ABCABC, the Pythagorean theorem yields AC2=122+162=400AC^2 = 12^2 + 16^2 = 400, so AC=20AC = 20. Next, in right triangle ADCADC, ACAC serves as the hypotenuse and AD=10AD = 10 is one leg. Solving for leg CDCD gives CD2=AC2−AD2=202−102=300CD^2 = AC^2 - AD^2 = 20^2 - 10^2 = 300, which simplifies to CD=300=103CD = \sqrt{300} = 10\sqrt{3}.

Adım Adım Çözüm

1
Draw diagonal ACAC to divide quadrilateral ABCDABCD into two right triangles, △ABC\triangle ABC and △ADC\triangle ADC, sharing hypotenuse ACAC.
Two right-angled triangles △ABC\triangle ABC (with right angle at BB) and △ADC\triangle ADC (with right angle at DD).
Diagonal ACAC acts as the hypotenuse for both right triangles.
2
Apply the Pythagorean Theorem to right triangle △ABC\triangle ABC to calculate the length of hypotenuse ACAC.
AC=AB2+BC2=122+162=144+256=400=20AC = \sqrt{AB^2 + BC^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 units.
In right triangle △ABC\triangle ABC, ABAB and BCBC are legs.
3
Apply the Pythagorean Theorem to right triangle △ADC\triangle ADC to solve for leg CDCD.
CD=AC2−AD2=202−102=400−100=300=103CD = \sqrt{AC^2 - AD^2} = \sqrt{20^2 - 10^2} = \sqrt{400 - 100} = \sqrt{300} = 10\sqrt{3} units.
In right triangle △ADC\triangle ADC, ACAC is the hypotenuse (2020) and ADAD is a leg (1010).

Anahtar Kavram

Multi-step applications of the Pythagorean Theorem using shared boundary hypotenuses.
Tahmini Süre:1m 15s
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Pythagorean Theorem and Special Right Triangles

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