Question:
In a triangle $ABC$, if $a = 13$, $b = 14$, $c = 15$, then the area of the triangle is:
In a triangle $ABC$, if $a = 13$, $b = 14$, $c = 15$, then the area of the triangle is:
Show Hint
A triangle with sides $13, 14, 15$ is a standard classic triangle in geometry problems; its area is always $84$, and its semi-perimeter is $21$. Remembering this saves valuable time.
Updated On: Jun 3, 2026
- 84
- 48
- 36
- 96
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Concept
The area of a triangle with known side lengths $a$, $b$, and $c$ can be calculated using Heron's Formula: $\Delta = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ is the semi-perimeter.
Step 2: Meaning
Here, the side lengths are $13$, $14$, and $15$.
Step 3: Analysis
First, calculate the semi-perimeter $s$: \[ s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \] Now, substitute $s$ and the sides into Heron's Formula: \[ \Delta = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \times 8 \times 7 \times 6} \] Simplify the terms inside the square root: \[ \Delta = \sqrt{(3 \times 7) \times (2^3) \times 7 \times (2 \times 3)} = \sqrt{7^2 \times 3^2 \times 2^4} \] \[ \Delta = 7 \times 3 \times 2^2 = 21 \times 4 = 84 \]
Step 4: Conclusion
The area of the triangle $ABC$ is exactly $84$ square units.
Final Answer: (A)
The area of a triangle with known side lengths $a$, $b$, and $c$ can be calculated using Heron's Formula: $\Delta = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ is the semi-perimeter.
Step 2: Meaning
Here, the side lengths are $13$, $14$, and $15$.
Step 3: Analysis
First, calculate the semi-perimeter $s$: \[ s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \] Now, substitute $s$ and the sides into Heron's Formula: \[ \Delta = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \times 8 \times 7 \times 6} \] Simplify the terms inside the square root: \[ \Delta = \sqrt{(3 \times 7) \times (2^3) \times 7 \times (2 \times 3)} = \sqrt{7^2 \times 3^2 \times 2^4} \] \[ \Delta = 7 \times 3 \times 2^2 = 21 \times 4 = 84 \]
Step 4: Conclusion
The area of the triangle $ABC$ is exactly $84$ square units.
Final Answer: (A)
Was this answer helpful?
0
0
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
View More Questions
Top AP EAPCET Some Properties of a Triangle Questions
- If $\Delta$ denotes the area of triangle $ABC$ and $s$ is its semi-perimeter, then
- In a triangle $ABC$, $C=90^\circ$. If $r$ is the inradius and $R$ is the circumradius of the triangle, then \[ 2(r+R)= \]
- The sides of a triangle are in the ratio \[ 1:\sqrt3:2. \] Then the angles opposite to these sides are in the ratio
- In \( \Delta ABC \), if \( A+B=120^{\circ} \), \( a=\sqrt{3}+1 \) and \( b=\sqrt{3}-1 \), then \( A:B= \)
- If the angles of a triangle are in the ratio 4: 1: 1, then the ratio between its largest side and its perimeter is
View More Questions
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
View More Questions