Question #185200
  1. the triangle ABC is right-angled with right angle at C. the height from the corner C divides the hypotenuse into two parts with one length 1 and the other 4. What is the circumference of the triangle.
  2. given a right triangle with catheter lengths a and b, determine and enter the length of the bisector at the right angle.
  3. a rectangle has a diagonal length d. the acute angle between the diagonals is x. given that cos (x) = p, determine and state the length of the shorter sides of the rectangle as a function of p
  4. Given a > 1, what is the biggest solution to: (1/x-1) ≥ a/(x+1)
  5. Of cos x = p, and pi/2<x<pi. Give a expression for tan x
1
Expert's answer
2021-05-07T08:59:56-0400

1)


a2=h2+12a^2=h^2+1^2

b2=h2+42b^2=h^2+4^2

52=a2+b25^2=a^2+b^2

52=1+b2+16+b25^2=1+b^2+16+b^2

2h2=2517=82h^2=25-17=8

h2=4h^2=4

h=2h=2

so a2=22+1=5a^2=2^2+1=5

a=2.236067977a=2.236067977


b2=22+16=20b^2=2^2+16=20

b=4.472135955b=4.472135955


circumference

=2.236067977+4.472135955+5=11.70820393=2.236067977+4.472135955+5=11.70820393


2)




BC=a,CA=b,AB=a2+b2BC=a, CA=b, AB=\sqrt{a^2+b^2}


CD is the angle bisector

let's draw square CEDF and CE=h

so,

BE=ah,CD=h2BE=a-h , CD=h\sqrt{2}


Now, ABC and DBE are similar triangles.

ba=hah\frac{b}{a}=\frac{h}{a-h}


h=aba+bh=\frac{ab}{a+b}


length of angle bisector is given by

CD=h2CD=h\sqrt{2}


CD=ab2a+bCD=\frac{ab\sqrt{2}}{a+b}



3)




shortest sides of the rectangle are AB and DC

using trigonometric ratio in triangle ABC.

<B=90°<B=90\degree

now,

cosx=adjacenthypotenusecosx=\frac{adjacent}{hypotenuse}


cosx=ABACcosx=\frac{AB}{AC}


cosx=ABdcosx=\frac{AB}{d}


now, p=ABdp=\frac{AB}{d}


AB=pdAB=pd


length of shortest side AB=p×dAB=p \times d




4)

a>1a>1


(1xax+1(\frac{1}{x}≥\frac{a}{x+1}


1xx+1x1a\frac{1}{x}-x+\frac{1}{x}-1≥a


xxx+1x1a\frac{x}{x}-x+\frac{1}{x}-1≥a


1x+1x1a1-x+\frac{1}{x}-1≥a


x+1xa-x+\frac{1}{x}≥a


1xxa\frac{1}{x}-x≥a


1x2xa\frac{1-x^2}{x}≥a



5)

cosx=pcos x=p


1secx=p\frac{1}{secx}=p


secx=1psec x=\frac{1}{p}


sec2x=(1p)2sec^2 x=(\frac{1}{p})^2


By identity

sec2xtan2x=1sec^2x-tan^2x=1


tan2x=sec2x1tan^2x=sec^2x-1


=1p21=\frac{1}{p^2}-1


=1p2p2=\frac{1-p^2}{p^2}


tan2x=1p2p2tan^2x=\frac{1-p^2}{p^2}


tanx=1p2p2tanx =\sqrt{\frac{1-p^2}{p^2}}


tanx=1p2ptanx=\frac{\sqrt{1-p^2}}{p}




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