The point where the internal angle bisectors of a triangle meet are called the incenter of a triangle. A triangle is a two-dimensional polygon that contains 3 vertices, 3 sides and 3 internal angles. Angle bisector divides the angle into two equal parts. 9th-grade students who are willing to learn the concept of geometry and measurement can get the useful details and prove for the statement bisectors of the angles of a triangle interest at a point.
Bisectors of the Angles of a Triangle Meet at a Point
This is proof for the bisectors of an angle of a triangle meet a point is the incenter of the circle.
Let us take a triangle XYZ. XI and YI bisects ∠YXZ and ∠XYZ respectively
We have to prove that IZ bisects ∠XZY
Draw IA ⊥ YZ, IB ⊥ XZ, IC ⊥ XY.
Proof:
| Statement | Reason |
|---|---|
| In ∆XIC and ∆XIB ∠CXI = ∠BXI ∠XCI = ∠XBI = 90° XI = XI |
XI bisects ∠YXZ Construction Common Side |
| ∆XIC ≅ ∆XIB | By AAS Criterion of Congruency |
| IC = IB | Corresponding parts of Congruent Triangles |
| Similarly, ∆YIC ≅ ∆YIA | |
| IC = IA | Corresponding parts of Congruent Triangles |
| IB = IA | From IC = IB and IC = IA |
| In ∆ZIA and ∆ZIB, IA = IB IZ = IZ ∠ZAO = ∠ZBO = 90° |
From IB = IA Common Side Construction |
| ∆ZIA ≅ ∆ZIB | By RHS criterion of congruency |
| ∠ZIA = ∠ZIB | Corresponding parts of Congruent Triangles |
NO bisects ∠XZY. (Proved)
Also, Check
- Point on the Bisector of an Anhttps://ccssanswers.com/gle
- Criteria for Congruency
- Congruency of Triangles
Incenter of a Triangle
The incenter of a triangle is defined as the point of intersection of all three interior angle bisectors of the triangle. In simple words, an incenter is a point where the internal bisectors of a triangle cross. This point is equidistant from the sides of a triangle, and it lies inside the triangle. The incenter of a triangle is the centre of its inscribed circle and it is the largest circle that exactly fit inside the triangle.
The formula to calculate incenter of a triangle is I = (\(\frac { ax₁+bx₂+cx₃ }{ a+b+c } \), \(\frac { ay₁+by₂+cy₃ }{ a+b+c } \)).
Here, (x₁, y₁), (x₂, y₂) and (x₃, y₃) are the coordinates of vertices of a triangle ABC and a, b and c are the lengths of its sides.
Incenter of a Triangle Properties
Below provided are the important properties of the triangle incenter.
- If I is the incenter of △ABC, then AE = AG, CG = CF and BE = BF.
- The sides of the triangle are tangents to the circle, thus EI = FI = GI and it is the inradii of the circle.
- If I is the incenter of △ABC, ∠BAI = ∠CAI, ∠BCI = ∠ACI and ∠ABI = ∠CBI by using angle bisector theorem.
- The triangle incenter always lies inside the triangle.
- If s is the semi perimeter of the triangle, r is the inradius of the triangle, then triangle area is the product of s, r.
Solved Examples on Bisectors of the Angles of a Triangle Meet at a Point
Example 1:
The coordinates of the incenter of the triangle ABC is formed by the points A(0, 3), B(3, 0), C(6, 3) is (x, y). Find (x, y).
Solution:
Given that,
Coordinates of triangle ABC are A(0, 3), B(3, 0), C(6, 3)
a = BC = √[(6 – 3)² + (3 – 0)²] = 4.24
b = AC = √[(6 – 0)² + (3 – 3)²] = 6
c = AB = √[(3 – 0)² + (0 – 3)²] = 4.24
Incenter of a triangle formula is I = (\(\frac { ax₁+bx₂+cx₃ }{ a+b+c } \), \(\frac { ay₁+by₂+cy₃ }{ a+b+c } \)).
I = (\(\frac { 4.24(0) + 6(3) + 4.24(6) }{ 4.24 + 6 + 4.24 } \), \(\frac { 4.24(3) + 6(0) + 4.24(3) }{ 4.24 + 6 + 4.24 } \)
= (\(\frac { 43.56 }{ 14.48 } \), \(\frac { 25.44 }{ 14.48 } \))
= (3, 1.75)
Therefore, (x, y) = (3, 1.75)
Example 2:
If I is the incenter of triangle ABC, ∠BAI = 37°, ∠CBI = 20°, ∠ACI = x° then find the value of x.
Solution:
Given that,
I is the incenter of the triangle
∠BAI = 37°, ∠CBI = 20°, ∠ACI = x°
AI, BI< CI are the angle bisectors.
So, ∠BAI + ∠CBI + ∠ ACI = 180°/2
37° + 20°+ x° = 90°
57° + x° = 90°
x°= 90° – 57°
= 33°
Therefore, x = 33°
Example 3:
David calculated the area of the triangular sheet as 50 sq ft. The perimeter of the sheet is 12 ft. If the circle is drawn inside the triangle such that it is touching every side of the triangle. Calculate the inradius of the triangle.
Solution:
Given that,
Area of the triangular sheet A = 50 sq ft
The perimeter of the sheet = 12 ft
Semiperimeter of the sheet s = 12/2 = 6 ft
The area of triangle A = sr
50 = 6 x r
r = 8.3
Therefore, the inradius of the triangle is 8.3 ft
Frequently Asked Question’s on Triangle Incenter
1. What is the term of the point where angles bisectors of a triangle meet?
The point where the angle bisectors of a triangle meet are called the inradius.
2. Can an incenter be outside a triangle?
No, the incenter of a triangle will always lie inside a triangle.
3. What Does Incenter Mean?
The incenter is the point where three bisectors of the interior angles of a triangle meet and it is the centre of the inscribed circle.