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
The sides of a triangle are in the ratio 6 : 4 : 3 and its perimeter is 104 cm. The length of the longest side (in cm) is:
An isosceles right-angled triangle has hypotenuse length as 10 units. What is the area of the triangle (in square units)?
Two circles of radii 16 cm and 4 cm, respectively, touch each other externally at Point A. PQ is the direct common tangent of these circles with centres C1 and C2, respectively. What is the length of PQ?
Let C be a circle with center O and AB be a chord of C such that the length of AB is equal to the radius of C. Let D be any point on the major arc of AB. Find ∠AOB and ∠ADB, respectively.
The centres of two circles are 84 cm apart. If the radii of these two circles are 38 cm and 26 cm, respectively, then which of the following options gives the length (in cm) of a direct common tangent of these two circles?
