Respuesta :
#1) The circumcenter may lie inside or outside the circle; all of the vertices of the triangle are equally distant from the circumcenter; the circumcenter is the center of the circumcircle.
#2) The incenter is equally distant from all of the sides of the triangle; the incircle around the incenter will touch all sides of the triangle; the biggest circle that can be drawn in a triangle is the incircle.
#3) The range of the possible lengths for n is 3<n<15.
Explanation:
#1 and 2 are properties of the terms, there is no explanation.
#3: The sum of any two sides of a triangle must be larger than the other side. This gives us the first inequality, (n+3)+(n+6)>3n-6.
Combine like terms:
2n+9>3n-6.
Subtract 2n from both sides:
2n+9-2n>3n-6-2n
9>n-6.
Add 6 to both sides:
9+6>n-6+6
15>n.
This gives us n<15.
The second inequality we have is (n+6)+(3n-6)>n+3.
Combine like terms:
4n>n+3.
Subtract n from both sides:
4n-n>n+3-n
3n>3.
Divide both sides by 3:
3n/3>3/3
n>1.
The third inequality we have is:
(n+3)+(3n-6)>n+6.
Combine like terms:
4n-3>n+6.
Subtract n from both sides:
4n-3-n>n+6-n
3n-3>6.
Add 3 to both sides:
3n-3+6>6+3
3n>9.
Divide both sides by 3:
3n/3>9/3
n>3.
We have the inequalities n<15, n>1 and n>3. If n is larger than 3, it automatically fits n>1, so we will narrow this to n<15 and n>3. This gives us the range 3<n<15.
#2) The incenter is equally distant from all of the sides of the triangle; the incircle around the incenter will touch all sides of the triangle; the biggest circle that can be drawn in a triangle is the incircle.
#3) The range of the possible lengths for n is 3<n<15.
Explanation:
#1 and 2 are properties of the terms, there is no explanation.
#3: The sum of any two sides of a triangle must be larger than the other side. This gives us the first inequality, (n+3)+(n+6)>3n-6.
Combine like terms:
2n+9>3n-6.
Subtract 2n from both sides:
2n+9-2n>3n-6-2n
9>n-6.
Add 6 to both sides:
9+6>n-6+6
15>n.
This gives us n<15.
The second inequality we have is (n+6)+(3n-6)>n+3.
Combine like terms:
4n>n+3.
Subtract n from both sides:
4n-n>n+3-n
3n>3.
Divide both sides by 3:
3n/3>3/3
n>1.
The third inequality we have is:
(n+3)+(3n-6)>n+6.
Combine like terms:
4n-3>n+6.
Subtract n from both sides:
4n-3-n>n+6-n
3n-3>6.
Add 3 to both sides:
3n-3+6>6+3
3n>9.
Divide both sides by 3:
3n/3>9/3
n>3.
We have the inequalities n<15, n>1 and n>3. If n is larger than 3, it automatically fits n>1, so we will narrow this to n<15 and n>3. This gives us the range 3<n<15.
1). The point at which perpendicular bisect each other is known as circumcenter.
2). The point at which angle bisectors intersect each other is known as incenter.
3). The range possible for n is [tex]\boxed{2<n<15}[/tex].
Further explanation:
(1)
An altitude is a line that is perpendicular to a side and passes through opposite vertex.
The point at which all the three perpendicular bisectors of a triangle intersect each is known as circumcenter.
The properties of the circumcenter is that the point may lie inside and outside of the triangle. It is point of intersection of altitudes. The vertices are at equal distance from the circumcenter.
(2)
The line that bisects the angle into two equal parts is known as the angle bisector.
The point at which all the three angle bisectors in a triangle intersect each is known as incenter of the triangle.
The property of the incenter is that the sides of triangle is at equal distance from the incenter.
(3)
The property of triangle is that the sum of two smaller sides must be greater than the larger side of the triangle.
Consider the larger side of the triangle is 3n-6.
The smaller sides are n+3 and n+6.
[tex]\begin{aligned}\left({n+3}\right)+\left({n+6}\right)&>3n-6\\2n+9&>3n-6\\9+6&>3n-2n\\15&>n\\\end{aligned}[/tex]
The side of triangle must be greater than zero.
[tex]\begin{aligned}3n-6&>0\\3n&>6\\n&>\frac{6}{3}\\n&>2\\\end{aligned}[/tex]
The range possible for n is [tex]\boxed{2<n<15}[/tex].
Learn more:
1. Learn more about inverse of the function https://brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function https://brainly.com/question/3412497
Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Triangles
Keywords: orthocenter, perpendicular, altitudes, point, triangle, intersect, centroid, circumcenter, bisectors, perpendicular bisectors, angles, angle bisectors, median, intersection, incenter, right angle triangle, equilateral triangle, obtuse, acute, range, length.
