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Hi, I've finished almost all of my Geometry homework besides the following couple:

 

1. In the diagram below, I is the incenter of \( \triangle \)ABC. We know that AB=14, BC=11, and CA=15. What is CE?

 

 

2. The side lengths of triangle ABC are 6,8, and 10. What is the inradius of ABC?

 

3. In the diagram below, I is the incenter of triangle ABC. The segment DE goes through I and is parallel to AC. If AE=4 and CD=3, what is DE? 

 

 

Thank you! 

 Nov 9, 2019
 #1
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2)

 

1. Semiperimeter of the triangle
The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles that it is given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

s = { p }/{ 2 } = { 24 }/{ 2 } = 12
2. The triangle area - from two legs
T = { ab }/{ 2 } = { 6 * 8 }/{ 2 } = 24
3. Inradius
An incircle of a triangle is a circle which is tangent to each side. An incircle center is called incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three angle bisectors. The product of the inradius and semiperimeter (half the perimeter) of a triangle is its area.

T = rs; r = { T }/{ s } = { 24 }/{ 12 } = 2

 Nov 9, 2019
 #2
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1. By similar triangles, CE = 22/3.

 

3. By similar triangles, DE = 8.5.

 Nov 10, 2019

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