Incircle Radius from Incenter Altitude
Understanding the Incenter and Inradius
The incenter of a triangle is the point where the angle bisectors intersect. It is equidistant from all three sides of the triangle.
The inradius ($r$) is the radius of the triangle's incircle, which is the circle inscribed within the triangle and tangent to all its sides. The distance from the incenter to any side of the triangle is equal to the inradius.
Identifying Key Information
- Incenter coordinates: $(3, 4)$
- Length of the altitude from the Incenter to side AB: $2 \text{ units}$
Relating Altitude to Inradius
The altitude from the incenter to a side (in this case, side AB) represents the perpendicular distance from the incenter to that side.
By definition, the inradius ($r$) is the perpendicular distance from the incenter to any side of the triangle.
Therefore, the length of the altitude from the incenter to side AB is equal to the inradius of the triangle.
Calculating the Inradius
Given that the length of the altitude from the Incenter to side AB is $2 \text{ units}$, the inradius ($r$) is directly determined:
$r = \text{Length of altitude from Incenter to side AB}$
$r = 2 \text{ units}$
The coordinates of the incenter $(3, 4)$ are extra information not required to determine the radius when the altitude length is provided.
Final Answer
The radius of the triangle's incircle is $2 \text{ units}$.
Similar Questions
Two right-angled triangular blocks, ABC and DEF, have ∠B = ∠E = 90°. If the lengths of the hypotenuses AC and DF are equal, and the sides AB and DE are equal, are the triangles congruent? If so, by what rule?
- A right-angled triangle ABC has legs AB=6 cm and BC=8 cm. An altitude BD is drawn from the vertex B to the hypotenuse AC. What is the length of the altitude BD?
The ratios of the perimeters of two similar triangles are expressed as 3 : 5. Together, their areas total 34 cm². What is the area of the larger triangle?
A right triangle has sides 3, 4, and 5. A smaller triangle is drawn inside it with its vertices on the sides of the larger triangle, such that it is similar to the larger triangle. If its perimeter is 6, what is its area?
A circle is circumscribed about a triangle. If one of the angles of the triangle is 120°, and the radius of the circumscribed circle is r, what is the length of the side opposite the 120° angle?
In triangle ABC, medians AD, BE, and CF intersect at the centroid G. What is the ratio of the area of triangle GAB to the area of triangle ABC?
In △ABC, a line segment DE is parallel to BC, with D on AB and E on AC. If the ratio of the area of △ADE to the area of the trapezoid DECB is 4 : 21, what is the ratio of AD to DB?
- In an urban project, triangular plots with side ratios 3:4:5 are allocated. If the perimeter is 60 m, find the area.
- In $\Delta ABC \sim \Delta XYZ$ and $AB = 4 \text{ cm}, BC = 6 \text{ cm}, XY = 8 \text{ cm}$, what is the length of YZ?
In a triangle ABC, medians AD and BE intersect at G. If the length of median AD is 12 cm, what is the length of the segment AG?
Important Questions from Triangles, Congruence and Similarity
G is the centroid of the equilateral triangle ABC. If AB = 8√ 3 cm, then the length of AG is equal to:
If sides of a triangle are 12 cm, 15 cm and 21 cm, then what is the inradius (in cm) of the triangle?
What is the area of quadrilateral ABCD?
It is given that ΔABC ~ ΔXYZ and Area ΔABC : Area ΔXYZ = 81 : 25. If AB = 18 cm, BC = 10 cm, CA = 15 cm, then what is the side XZ (in cm)?
Sides of two similar triangles are in the ratio 4 ∶ 9. Area of these triangles are in the ratio:
