Triangle Inscribed Circle Radius Formula
The radius of a circle inscribed within a triangle (also known as the inradius) can be determined using the triangle's area and side lengths. The key formula connecting these elements is:
$ A = rs $
- A represents the area of the triangle.
- r represents the radius of the inscribed circle (inradius).
- s represents the semi-perimeter of the triangle.
Calculating the Semi-perimeter
The semi-perimeter ($s$) is half the sum of the lengths of the triangle's sides ($a$, $b$, $c$).
$ s = \frac{a+b+c}{2} $
Deriving the Inradius Formula
Substitute the expression for the semi-perimeter ($s$) into the area formula ($A = rs$):
$ A = r \left( \frac{a+b+c}{2} \right) $
To find the radius ($r$), rearrange the equation:
$ r = \frac{A}{\left( \frac{a+b+c}{2} \right)} $
Simplify the expression:
$ r = \frac{2A}{a+b+c} $
Conclusion
Therefore, the radius of the inscribed circle in triangle ABC is given by the formula $\frac{2A}{a+b+c}$. This matches Option B.
Similar Questions
- The number of triangles formed by diagonals from one vertex of an octagon is:
- The sum of all interior angles of a 20-sided polygon is:
- If the measure of an exterior angle of a regular polygon is $20^\circ$, determine the total number of its sides.
- What is the minimum distance between centers of two circles having radii 5 cm and 3 cm such that exactly three common tangents exist?
- In a circle, angle subtended by chord AB at the center is $80^\circ$. What is the measure of angle $\angle ACB$, where C lies on the circle?
- In a circle with center O, two chords, AB and CD, cross each other at right angles. if the distance from the center O to chord AB is 3 cm, while the distance from O to chord CD is 4 cm. What is the radius of the circle?
- If the angle subtended by a chord at the center is $150^\circ$, what is the angle subtended at a point on the circle in the same segment?
- In a triangle ABC, if the angle bisectors of $\angle B$ and $\angle C$ intersect at point I, and $\angle BIC = 110^\circ$, what is the measure of $\angle A$?
- In triangle PQR, ST is parallel to QR. If PS=5cm, SQ=10cm, and PT=6cm, find TR.
- In a triangle ABC, side BC is extended to a point D. The exterior angle at C is 105°. The interior angles at A and B are in the ratio 2:5. What is the measure of the largest interior angle of the triangle?
Important Questions from Geometry
ABCDEF is a regular hexagon. Side of the hexagon is 36 cm. What is the area of the triangle AOB ?
If ∆ABC ~ ∆DEF, and BC = 4 cm, EF = 5 cm and the area of triangle ABC = 80 cm 2, then the area of the triangle DEF is:
If Δ ABC is right angled at B, AB = 12 cm and ∠CAB = 60°, determine the length of BC.
If ΔABC and ΔDEF are congruent triangles, then which of the following is FALSE?
D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F. then find the ratio of areas of ΔDEF and ΔCBF.
