What Is the Blue Area? This Was Asked on the SAT

Comments · 13

• XI

  @Xiassen · 5 days ago

  A quadrilateral; next question.

  1

• TH

  @theohopkins1903 · 5 days ago

  The small circle just a prop to hold the blue square in position. Ignore it!

• KA

  @KailashChandraGUPTA-s1y · 16 hours ago (edited)

  Let the lowest point if the blue square be A,upper point of its side slanting to right be B,mid point of AB be C,left lower corner if the square 18 be D ,it&apos;s lower right corner be I.Also let the point where square 8 touches &nbsp;the side of &nbsp;the square18 &nbsp;be F ant it&apos;s left lower point be E,also let the point where square 2 &nbsp;meets &nbsp;the square 8 &nbsp;at lower left be G. Let the length AB &nbsp;be x. Since the lengths of the sides of squares 18,8 and 2 are respectively 3√2,2√2 and 1√2,vertical distance of C from B is the same &nbsp;as the vertical distance of A from C( each being equal to 3√2).So distance AC=(1/2)x.Drop perpendiculer &nbsp;from B onDI which meets it as point H.Let AD=a.Then AH=AD+DH=AD+CE+FG= a+(3√2-2√2)+((1/2){(2/√2-1√2)}=a+ 1√2+(1/2)(1√2)=a+ 1 .5√2<br>.Now AB^2=AH^2+HB^2= (a+ 1.5√2)^2+(6√2)^2(since BH is equal to sum of the sides of the squares 18,8 and 2=3√2+2√2+1√2=6√2).So a^2+3a√2+4.5+(6√2)^2=a^2+3a√2+4.5+72........Y.Also AC^2= AD^2+DC^2or{ (1/2)x}^2=a^2+(3√2)^2 or ( 1/4)x^2= a^2+18 or x^2= 4a^2+72......Z.From equations Y and Z,a^2+3√2a.4.5+72=4a^2+72 or 3a^2-3a√2-4.5=0 .So x=1.5√2 or -.5√2.Discarding negative value a=1.5√2.AB^2=AH^+HB^or x^2= (1.5√2+1.5√2)^2+(6√2)^2=18+72=90 &nbsp;which is also the area of blue square.

• PA

  @paparmar · 5 days ago (edited)

  @<a href="https://www.youtube.com/watch?v=XpiC5gUhvg4&amp;t=135">2:15</a>: Once you establish the heights of the 3 similar right triangles are in the ratio 1:2:3, you know the ratio of their hypotenuses is also that. &nbsp;The tilted side of the blue square (length s) is comprised of the 3 hypotenuses; the middle triangle&apos;s hypotenuse accounts for one-third of the total of the hypotenuses (2/6 = 1/3). &nbsp;Thus s = 3*(middle hypotenuse) or s^2 = 9*(middle hypotenuse)^2. &nbsp;By Pythagoras, (middle hypotenuse)^2 = 10. Hence area square = s^2 = 9*10 = 90. &nbsp;Note you avoid dealing with any more square roots by going straight to the area s^2.

• מש

  @משהנאון-י6מ · 3 days ago

  S=900

• MO

  @monroeclewis1973 · 6 days ago

  How long were SAT examinees given to answer?

  1

• ST

  @SteveBull-tg8mi · 5 days ago

  I doubt this is a question on the Sat.

  2

• SP

  @SpaceManSpiff26 · 5 days ago

  better yet: What is the circumference of the circle?