Question #197487

14. The perimeter of the right-angled triangle ABC is 80 cm. Its base is 6x and one of the side ribs x². Find x.


1
Expert's answer
2021-06-02T08:43:11-0400

If one of the side ribs, which equals x² is hypotenuse, then the other rib length is: b2=(x2)2(6x)2=x436x2.b^2 = (x^2)^2 - (6x)^2 = x^4 - 36x^2.

b=x436x2b = \sqrt{x^4 - 36x^2}

The perimeter of this triangle is:

P=x2+6x+x436x2=80,P = x^2 + 6x + \sqrt{x^4 - 36x^2} = 80,

x436x2=80x26x,\sqrt{x^4 - 36x^2} = 80 - x^2 - 6x,

x436x2=6,40080x2480x80x2+x4+6x3480x+6x3+36x2,x^4 - 36x^2 = 6,400 - 80x^2 - 480x - 80x^2 + x^4 + 6x^3 - 480x + 6x^3 + 36x^2,

12x388x2960x+6,400=0,12x^3 - 88x^2 - 960x + 6,400 = 0,

3x322x2240x+1,600=0,3x^3 - 22x^2 - 240x + 1,600 = 0,

(3x330x2)+(8x2240x+1,600)=0,(3x^3 - 30x^2) + (8x^2- 240x + 1,600) = 0,

3x2(x10)+8(x230x+200)=0,3x^2(x - 10) + 8(x^2 - 30x + 200) = 0,

3x2(x10)+8(x10)(x20)=0,3x^2(x - 10) + 8(x - 10)(x - 20) = 0,

(x10)(3x2+8x160)=0,(x - 10)(3x^2 + 8x - 160) = 0,

So, x1 = 10 is one of the roots of this equation.

D = 64 + 1920 = 1984.

x2=8+19846=6.09,x2 = \frac{-8 + \sqrt{1984}}{6} = 6.09,

x3=819846=8.76,x3 = \frac{-8 - \sqrt{1984}}{6} = -8.76, which is not possible in our case.


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