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Intern  Joined: 17 Jul 2013
Posts: 47
GMAT 1: 710 Q49 V38 GRE 1: Q166 V160 GPA: 3.74
XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the h  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 59% (01:54) correct 41% (02:27) wrong based on 219 sessions

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XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the hypotenuse. A circle with center O and radius x has been inscribed in the triangle. What is the value of x?

A) 1.8
B) 2
C) 2.4
D) 3
E) 4
Manager  Joined: 22 Feb 2009
Posts: 156
Re: XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the h  [#permalink]

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megatron13 wrote:
XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the hypotenuse. A circle with center O and radius x has been inscribed in the triangle. What is the value of x?

A) 1.8
B) 2
C) 2.4
D) 3
E) 4

Draw the triangle, connect OX, OY, OZ
Hypotenuse = 10
Area of XYZ = 8*6/2 = 24
Area of OXZ = x*10/2 = 5x
Area of OXY = x*8/2 = 4x
Area of OYZ = x*6/2 = 3x
Area of OXZ + Area of OXY + Area of OYZ = Area of XYZ -> 5x + 4x + 3x = 24 ---> x = 2
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Senior Manager  Joined: 13 Jun 2013
Posts: 266
Re: XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the h  [#permalink]

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megatron13 wrote:
XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the hypotenuse. A circle with center O and radius x has been inscribed in the triangle. What is the value of x?

A) 1.8
B) 2
C) 2.4
D) 3
E) 4

Attachments

File comment: solution finalsol.jpg [ 41.76 KiB | Viewed 26065 times ]

Intern  Joined: 19 Jul 2013
Posts: 38
Re: XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the h  [#permalink]

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megatron13 wrote:
XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the hypotenuse. A circle with center O and radius x has been inscribed in the triangle. What is the value of x?

A) 1.8
B) 2
C) 2.4
D) 3
E) 4

Easier solution.
Draw the diagram. Let the circle touch XY at P, YZ at Q and XZ at R. Then PY = YQ = radius of the circle(x). Now remember all the sides of the triangle are like tangents to the circle and both tangents from any external point to a circle are equal in size. So XP = XR and RZ = QZ. Also XP = 8-x and QZ = 6-x. So (8-x)+(6-x)=XP+QZ=XR+RZ=XZ=10. So x=2.
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Re: XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the h  [#permalink]

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1
area of triangle =1/2bh

Another formula is , r s , r= inradius of triangle and s= semiperimeter of triangle.
here sides 6,8and 10.
area of triangle =1/2 *6*8=24
semiperimeter ,s=>12
rs=24=> r *12=24
r=2.
Option B is sufficient.

Here some lists of area of triangles formula.
A=1/2bh

A=1/2ab sinC, a and b are two sides , both included angle SinC

A=abc /4R ,=> R= circumradius of triangle.

A=rs , r is inradius of triangle, s is semiperimeter of triangle
Intern  Joined: 12 Jul 2015
Posts: 25
Location: United States
Concentration: Strategy, General Management
WE: General Management (Pharmaceuticals and Biotech)
Re: XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the h  [#permalink]

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2
The 3 sides of a right angle triangle = 6,8,10

Radius of a circle inscribed in the right triangle r = (a+b-c)/2 where c is the hyp and a and b are other two sides.

On substituting: (6+8-10)/2
r = 2

Its a time saver!
Intern  Joined: 30 Nov 2015
Posts: 7
Re: XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the h  [#permalink]

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dayavenket wrote:
The 3 sides of a right angle triangle = 6,8,10

Radius of a circle inscribed in the right triangle r = (a+b-c)/2 where c is the hyp and a and b are other two sides.

On substituting: (6+8-10)/2
r = 2

Its a time saver!

is this a legit formula?
Intern  Joined: 12 Jul 2015
Posts: 25
Location: United States
Concentration: Strategy, General Management
WE: General Management (Pharmaceuticals and Biotech)
Re: XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the h  [#permalink]

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1
daisuke123 wrote:
dayavenket wrote:
The 3 sides of a right angle triangle = 6,8,10

Radius of a circle inscribed in the right triangle r = (a+b-c)/2 where c is the hyp and a and b are other two sides.

On substituting: (6+8-10)/2
r = 2

Its a time saver!

is this a legit formula?

Yes, You can derive this equation easily. Find enclosed the picture.

If r be the radius of the inscribed circle, then the red length on the vertical edge is given by a−r; similarly the blue length on the horizontal edge is given by b−r.

Using the property that tangential distances are equal we get a−r+b−r=c.

=> a+b−2r = c or r=(a+b−c)/2

Hope this helps! Attachments

File comment: Right triangle right triangle.png [ 9.57 KiB | Viewed 23726 times ]

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Re: XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the h  [#permalink]

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Simply use this formula:
Area of triangle = r.s ( r is the radius of the inscribed circle and s is the sum of sides /2 )
so here 1/2 b h = r.s => 1/2*8*6 = x* (6+8+10)/2
=> x = 2 (answer) Re: XYZ is a right angled triangle with YZ = 6cm and XY = 8cm. XZ is the h   [#permalink] 18 Aug 2018, 21:41
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