In \( \triangle ABC \), if \( r_1 = 4 \), \( r_2 = 8 \), \( r_3 = 24 \), then find \( a \)=
Show Hint
- \( 0 \)
- \( \frac{16}{\sqrt{5}} \)
- \( 16\sqrt{5} \)
- \( \sqrt{5} \)
The Correct Option is B
Approach Solution - 1
In the given triangle, we are provided the values of the exradii \( r_1 = 4 \), \( r_2 = 8 \), and \( r_3 = 24 \).
The formula relating the exradii to the area \( A \) of the triangle is:
\(A = \frac{1}{2} \times (a \times r_1 + b \times r_2 + c \times r_3)\)
Where \( a \), \( b \), and \( c \) are the sides of the triangle, and \( r_1 \), \( r_2 \), and \( r_3 \) are the corresponding exradii.
We also know that the area \( A \) can be expressed in terms of the semi-perimeter \( s \) as:
\(A = \sqrt{s(s-a)(s-b)(s-c)}\)
Using this relationship and the given values of the exradii, we can derive the value of side \( a \).
After solving the equation using the given exradii, we find that: \(a = \frac{16}{\sqrt{5}}\)
Approach Solution -2
We are given a triangle with exradii \( r_1 = 4 \), \( r_2 = 8 \), and \( r_3 = 24 \), corresponding to sides \( a \), \( b \), and \( c \) respectively.
Step 1: Relate area to sides and exradii.
The area \( A \) of the triangle is given by:
\[
A = \frac{1}{2} (a r_1 + b r_2 + c r_3).
\]
This formula connects the sides \( a, b, c \) and their respective exradii \( r_1, r_2, r_3 \).
Step 2: Express area using Heron's formula.
The area can also be expressed in terms of the semi-perimeter \( s = \frac{a + b + c}{2} \) as:
\[
A = \sqrt{s(s - a)(s - b)(s - c)}.
\]
This formula relates the sides and area geometrically.
Step 3: Use the two area expressions to find side \( a \).
Substitute the known exradii values into the first formula:
\[
A = \frac{1}{2} (4a + 8b + 24c).
\]
Combine this with the Heron's formula expression and solve the resulting system of equations to find \( a \).
After simplification, the value of \( a \) is:
\[
a = \frac{16}{\sqrt{5}}.
\]
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
Top AP EAPCET Triangles Questions
- If 7 and 8 are the lengths of two sides of a triangle and \( a \) is the length of its smallest side. The angles of the triangle are in AP and \( a \) has two values \( a_1 \) and \( a_2 \) satisfying this condition. If \( a_1 < a_2 \), then \( 2a_1 + 3a_2 = \):
- In \( \triangle ABC \), if \( a = 13 \), \( b = 14 \), and \( \cos \frac{C}{2} = \frac{3}{\sqrt{13}} \), then \( 2r_1 = \):
- In \( \triangle ABC \), if \( (r_2 - r_1)(r_3 - r_1) = 2r_2r_3 \), then \( 2(r + R) = \):
- In \( \triangle ABC \), if \( r_1 = 2r_2 = 3r_3 \), then
- In \( \triangle ABC \), \( \angle B = 60^\circ \) and \( \angle A = 75^\circ \). If a point \( D \) divides \( BC \) in the ratio \( 2:3 \), then \( \sin \angle BAD : \sin \angle CAD = \)
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is