Circle
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The correct answer is C
Huỳnh Anh Phương selected this answer.
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Name the points as shown
Draw \(OH\perp BC\),then H is the midpoint of BC.So BH = 1 cm
Quadrilateral ABHK has 3 right angles at A,B and H
=> ABHK is a rectangle => HK = AB = 2 cm
Let r be the radius of the circle,so OH = KH - OK = 2 - r (cm)
\(\Delta OHB\) right at H has : OH2 + HB2 = OB2 (Pythagoras theorem)
\(\Rightarrow\left(2-r\right)^2+1^2=r^2\Rightarrow4-4r+r^2+1-r^2=0\)
Carter selected this answer.
\(\Rightarrow5-4r=0\Rightarrow r=\dfrac{5}{4}\) -
FA KAKALOTS 03/02/2018 at 12:37
Name the points as shown
Draw OH⊥BC
,then H is the midpoint of BC.So BH = 1 cm
Quadrilateral ABHK has 3 right angles at A,B and H
=> ABHK is a rectangle => HK = AB = 2 cm
Let r be the radius of the circle,so OH = KH - OK = 2 - r (cm)
ΔOHB
right at H has : OH2 + HB2 = OB2 (Pythagoras theorem)
⇒(2−r)2+12=r2⇒4−4r+r2+1−r2=0
⇒5−4r=0⇒r=54
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mathlove 16/03/2017 at 19:06
We have \(AB=AM+BM=10+6=16;BC=BN+CN=6+x;CA=CP+AP=x+10\).
The triangle ABC has his edges \(a=10+6=16;b=6+x;c=x+10\) . Call \(p\) is the halp perimeter of \(ABC\) then \(p-a=10,p-b=6,p-c=x\) and the area of ABC is
\(\sqrt{p\left(p-a\right)\left(p-b\right)\left(p-c\right)}=\sqrt{60x\left(16+x\right)}\) .
By the assumptions we have \(60x\left(x+16\right)=\left(120\sqrt{3}\right)^2\Leftrightarrow x^2+16x-720=0\Leftrightarrow x\in\left\{20;-36\right\}\).
So, \(x=20;p=36\), the perimeter of ABC is 72.
Suppose CH is the height of CAB, then \(CH=\dfrac{2.120\sqrt{3}}{16}=15.\) The triangle CHA has
\(CHA=90^0;CH=15;CA=30\Rightarrow A=60^0\).
The measure of angle A is \(60^0\) .
Selected by MathYouLike -
FA KAKALOTS 03/02/2018 at 12:37
We have AB=AM+BM=10+6=16;BC=BN+CN=6+x;CA=CP+AP=x+10
.
The triangle ABC has his edges a=10+6=16;b=6+x;c=x+10
. Call p is the halp perimeter of ABC then p−a=10,p−b=6,p−c=x
and the area of ABC is
√p(p−a)(p−b)(p−c)=√60x(16+x)
.
By the assumptions we have 60x(x+16)=(120√3)2⇔x2+16x−720=0⇔x∈{20;−36}
.
So, x=20;p=36
, the perimeter of ABC is 72.
Suppose CH is the height of CAB, then CH=2.120√316=15.
The triangle CHA has
CHA=900;CH=15;CA=30⇒A=600
.
The measure of angle A is 600
.