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Zorluk: OrtaTriangles and Pythagorean Theorem

A right-angled triangular bracket is used to support a shelf. The longest side (hypotenuse) of the bracket has a length represented by the expression 2x+12x + 1 inches, and the horizontal leg has a length represented by 3x−63x - 6 inches. If x=6x = 6, what is the length of the vertical leg?

  1. A
    1
  2. 5Cevap
  3. C
    25
  4. D
    313

Cevap

The vertical leg has a length of 55 inches.
After substituting x=6x = 6, the hypotenuse is 1313 inches and the horizontal leg is 1212 inches. Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we get b2+122=132b^2 + 12^2 = 13^2. Solving for bb yields b2=25b^2 = 25, which means b=5b = 5.

Adım Adım Çözüm

1
Substitute x=6x = 6 into the expression for the hypotenuse.
2(6)+1=12+1=13 inches2(6) + 1 = 12 + 1 = 13\text{ inches}
To find the numerical length of the longest side.
2
Substitute x=6x = 6 into the expression for the horizontal leg.
3(6)−6=18−6=12 inches3(6) - 6 = 18 - 6 = 12\text{ inches}
To find the numerical length of the given leg.
3
Apply the Pythagorean theorem to find the vertical leg (let's call it bb).
b2+122=132b^2 + 12^2 = 13^2
The Pythagorean theorem relates the three sides of a right triangle: a2+b2=c2a^2 + b^2 = c^2.
4
Solve for bb.
b2+144=169⇒b2=169−144=25⇒b=5 inchesb^2 + 144 = 169 \Rightarrow b^2 = 169 - 144 = 25 \Rightarrow b = 5\text{ inches}
To calculate the final missing side length.

Anahtar Kavram

To find a missing leg in a right triangle when given the hypotenuse and the other leg, substitute the known values into the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) and solve for the unknown leg by subtracting the squares and taking the square root.

Alternatif Yöntem

Recognize that 55, 1212, and 1313 form a common Pythagorean triple. Once the hypotenuse (1313) and one leg (1212) are calculated, the other leg must immediately be 55.
Tahmini Süre:1m 15s
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Konular

Triangles and Pythagorean Theorem

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