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
- 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?
- 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?
- If in $\Delta LMN$ and $\Delta OPQ$, $\angle L = \angle O$, $LM = OP$, and $\angle M = \angle P$, which rule proves $\Delta LMN \cong \Delta OPQ$?
- A triangle with all its angles less than $90^\circ$ has its Orthocenter located where?
- In triangle ABC, DE is drawn parallel to BC, intersecting AB and AC at D and E respectively. If AD=3, DB=6, and AE=4, find EC.
- Triangle PQR has vertices P(1, 1), Q(4, 1), and R(1, 5). Triangle STU has vertices S(6, 2), T(9, 2), and U(6, 6). Are these two triangles congruent?
- Two triangles, $\Delta PQR$ and $\Delta STU$, have $PQ=ST$, $PR=SU$, and $\angle QPR = \angle TSU$. By what rule are they congruent?
- If in triangle ABC, $DE \parallel BC$ and $AD/DB = \frac{1}{2}$, and $AE = 5\text{ cm}$, then $EC$ is:
- For a triangle with one interior angle greater than 90°, where is its Orthocenter located?
Important Questions from Triangles, Congruence and Similarity
The radius of the circumcircle of an equilateral triangle of √3 unit side, is:
If the ratio of the angles of a triangle is 3 : 5 : 7, find the value of the largest angle.
A. 36°
B. 60°
C. 84°
D. 15°
If ΔABC and Δ PQR are similar and \(\rm\frac{BC}{QR} = \frac{1}{3}\) , find \(\rm\frac{ar(\Delta PQR)} {ar(\Delta BCA)}\)
If the angles of a triangle are in the ratio of 2 : 5 : 8, then find the value of the smallest angle.
ABCD is a parallelogram. Side BC is produced to E such that BC = CE. Join AE which intersects side CD at P. The area of triangle ABE is:
