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# ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is

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Re: ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is [#permalink]
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OA: B
It is a right angled triangle and 6,8 and 10 are Pythagorean triplet.
So AC=10
Radius of Incircle of Right angled triangle is given $$r =\frac{(a+b−c)}{2}$$
Where c is hypotenuse, a and b are other two sides.

Reason for AE=AD and EC=CF: length of tangents drawn from an external point to a circle are equal

here putting c = 10, a=6 , b= 8, we get $$r =\frac{(6+8−10)}{2}=2$$
OA=B
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Re: ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is [#permalink]
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005ashok wrote:

ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is hypotenuse. A circle with centre O and radius x has been inscribed. What is the value of x.
A. 2.4 cm
B. 2 cm
C. 3.6 cm
D. 4 cm
E. 3 cm
Attachment:
1796929.png

Option B it is.
I have a different way of solving it. Please give kudos if it seems interesting to you.

Its obvious AC is 10. Now, if a draw a perpendicular from B on AC (Let the length of perpendicular be P), then that line will coincide with the diameter of incircle(with r radius) extended to point B.
Now, equate the area of triangle ABC

1/2*AB*BC = 1/2*AC*P
From this we get P = 4.8

P can also be written as r + r*root2.

Hence, when r = 2, then 2 + 2*root2 gives us 2 + 2*1.4 = 2 + 2.8 = 4.8
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Re: ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is [#permalink]
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This may be a useful shortcut:-
AP = 8-r; And since we know that length of tangents from a single point are equal, hence, AM = 8-r
Similarly, CM = 6-r
Also by Pythagoras theorem, AC = 10
So AP + CM = AC
=> 8-r + 6-r = 10
Hence r = 2. et voilà
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Re: ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is [#permalink]
Knowing the in-radius rule for this sort of thing can be helpful, but I wonder if simply trying to visualize that last step may be good, too.

3 of the answer choices can be quite quickly eliminated off the bat.
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Re: ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is [#permalink]
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First, we have a multiple of a 3-4-5 right triangle ——-> hypotenuse = AC = 10 cm

Since we are given the radius = r from point O to M at the hypotenuse, it becomes an exercise of setting the hypotenuse equal to 2 expressions

Rule: When the radius of a circle is drawn to the point of tangency, it creates a 90 degree angle with the tangent line

Thus the radius OP is perpendicular to side AB

As well as the radius ON is perpendicular to side BC

This creates a square OPBN of side r

The remaining length from point N to C = NC = 6 - r

The remaining length from point P to A = PA = 8 - r

(2) rule: when 2 lines are drawn from the same exterior point and are tangent to the same circle, the distance from the exterior point to the point of tangency will be equal

Since we know that ON is the radius of the inscribed circle, OM will be drawn perpendicular to the hypotenuse.

The inscribed circle is tangent to the Right Triangle at Points: M , P, and N

(From exterior point C)

CM and NC will be equal tangent lines drawn from the same exterior point

We found the value of NC above

NC = 6 - r = CM

(From exterior point A)

DA and AM will be equal tangent lines drawn from the same exterior point

DA = 8 - r = AM

Finally, the hypotenuse = AM + CM = 10

Which means:

(6 - r) + (8 - r) = 10

14 - 2r = 10

4 = 2r

r = 2

Posted from my mobile device
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Re: ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is [#permalink]
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This question was part of 10th board exam in Indian CBSE board and I somehow remebered it.Nostalgia
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ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is [#permalink]
005ashok wrote:

ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is hypotenuse. A circle with centre O and radius x has been inscribed. What is the value of x.

A. 2.4 cm
B. 2 cm
C. 3.6 cm
D. 4 cm
E. 3 cm

Attachment:
1796929.png

By far, the easiest method to answer this is via approximation. Comparing BC to BN, for example by putting your fingers or pen or paper to the screen, copying the length of BN and placing it repeatedly along BC, BN seems to be almost exactly one third of BC. Answer B.

The problem though is that this is not actually a GMAT question - so it's not really to scale. If this were a real GMAT question, this would be the technique to use.
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Re: ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is [#permalink]
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Re: ABC is a right angled triangle with BC = 6 cm and AB = 8 cm. AC is [#permalink]
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