Geometry Problem
Click the figure below to see the complete problem 459 about Right triangle, Squares, Distance, Measurement.
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Complete Problem 459
Level: High School, SAT Prep, College geometry
Friday, May 21, 2010
Problem 459: Right triangle, Squares, Distance, Measurement
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cos(GAD)= cos(GAC+CAD)=cos(GAC+45)=cos(GAC)cos(45)-sin(GAC)sin(45). But cos(GAC)=c/b & sin(GAC)=a/b hence cos(GAD)=(c-a)/(b√2). From Tr. GAD using the co-sine rule, we’ve: x^2 = GA^2 +AD^2 -2GA*AD*cos(GAD) = (c+a)^2+2b^2-2*(c+a)( b√2)(c-a)/(b√2)= (c+a)^2 + 2(c^2+a^2) -2*(c+a)(c-a)= c^2 + 2ac + a^2 + 4a^2 = 5a^2 + c^2 + 2ac
ReplyDeleteAjit
Let m projection of AB over AC.
ReplyDeleteLet n projection of BC over AC.
Let h altitude over hypotenuse AC.
Then (T.altitude) h = sqrt (mn)
Then (T.legs) m = c^2/sqrt(a^2+c^2)
n = a^2/sqrt(a^2+c^2)
Let H point of intersection of parallel to AC from G and extension of side DC from C. Then GHD is right triangle on H.
We've the next relations:
CD = AC = m + n = sqrt(a^2 +c^2) (Pt. Tri.ABC)
CH = h + n
GH = h - n
HD = CH + CD = h + 2n + m
Then:
x^2=GH^2 + HD^2 = (h-n)^2 + (h + 2n + m)^2 =
= ... = 6mn + 2*sqrt(mn)*(m+n) + 5n^2 + m^2 = ... =
= 5a^2 + 2ac + c^2
MIGUE.
Let H point on line GF, and HF=GF. Then CG=CF and angle GCH=90.
ReplyDeleteOn triangle CDG and triangle CAH, CD=CA and CG=CH.
angle DCG=90+angle DCH, angle ACH=90+angle DCH.
So, triangle CDG==triangle CAH ---> then, AH=DG=x
On triangle AGH, angle G=90, AG=a+c, GH=2a, AH=x.
Therefore, x^2=(a+c)^2+(2a)^2 ---> x^2 = 5a^2+c^2+2ac
@ third comment by Anonymous:
ReplyDeleteCG will never be equal to CF as CG is the diagonal of the square and CF the side
Pls correct this comment
By boxing the largest square ACDE its possible to make a new right triangle with x as its hypotenuse that's easy to compute:
ReplyDeleteFirst rotate and translate triangle ABC so that its hypotenuse lies on CD and its new vertex is H.
Then extend the DH side so that it meets FG at I forming a new right triangle.
DI = a + a
GI = a + c
x^2 = DI^2 + GI^2 = (2a)^2 (a+c)^2 = 5a^2 + 2ac + c^2
Extend GF to H such that FH = a
ReplyDeleteThen Tr.s DCG and ACH are congruent SAS and x^2 = AH^2 = (a+c)^2 + 4a^2 from which the result follows
Sumith Peiris
Moratuwa
Sri Lanka
Let AC=b. Extend DC to P such that C is the mid-point i.e. DC=CP=b. Connect FG to P and observe that FCP is congruent to CBA (SAS). Hence FP=c.
ReplyDeleteConstruct a line from D parallel to CF such that it meets GF extended at Q.
Using mid-point theorem for the right tringle DPQ, it can be easily seen that DQ=2.CF=2a
FQ=FP=c.
As DQG is a right triangle with DQ=2a, GQ=GF+FQ=a+c,=> x^2=(2a)^2+(a+c)^2
The result follows.