Weird Triangle

By Heron's Formula we have the area of triangle as $\sqrt {s(s-a)(s-b)(s-c)}$ where $a,b,c$ are the side lengths and $s$ is the semi-perimeter which is equal to $\frac{15+41+52}{2}$ in this case. Plugging in the values gives us $\sqrt {54(54-15)(54-41)(54-52)}=\boxed {234}$

Area=sqrt(s(s-a)(s-b)(s-c)) ; s=(a+b+c)/2 a=15,b=41,c=52 and s=54 ; Area=sqrt(54(54-15)(54-41)(54-52)) Area=sqrt(54(39)(13)(2)) Area=sqrt(54756) Area=234

The area formula for every triangle is known and is $A=\dfrac{height*base}{2}$ . So, first of all there is necessary to calculate the height of the triangle, for that we write the next two equations system: $(52-x)^2+(h)^2=(41)^2; (x)^2+(h)^2=(15)^2$ . Now we procede to calculate the systems equations by the reduction method: $(52-x)^2+(h)^2=(41)^2$ . $-(x)^2-(h)^2=-(15)^2 \\ (52-x)^2- (x)^2=(41)^2-(15)^2 \\ \Rightarrow (52)^2-2*52*x+(x)^2 -(x)^2 \\ =1681-225 \\ =1456 \\ \Rightarrow -2*52*x=1456-(52)^2=-1248 \\ \Rightarrow x=1248/2*52=12$ .

Now we substitute the x in one of the two equations to know the value of h. $(h)^2=(15)^2-(12)^2=81 \\ \Rightarrow h=\sqrt{81}=9$

Hence the area of the triangle is $A=\dfrac{52*9}{2}=234.\square$

The area is $\sqrt{ S * (S-a) * (S-b) * (S-c) }$ $S=\frac{(a+b+c)}{2}$ and a=15 ,b=41 and c=52

a,b,c be the side length

p = $(a+b+c)/2\\ = (15+41+52)/2\\ = 54$

Area = $\sqrt{p\times(p-a)\times(p-b)\times(p-c)}\\ = \sqrt{54\times(54-15)\times(54-41)\times(54-52)} \\ = \sqrt{54756} \\ = 234.$

Given sides are $15,41,52$ . Let us take these sides as $a,b,c$ respectively and $s$ be the semi-perimeter of the triangle, i.e, half the perimeter of the triangle.

We shall use here Heron's formula which states that, Area $=\sqrt{s(s-a)(s-b)(s-c)}$ for a triangle of sides $a,b,c$ and semi-perimeter $s$ .

Perimeter of this triangle $=15+41+52=108$ . So, $s=\frac{108}{2}=\boxed{54}$

Area $=\sqrt{54(54-15)(54-41)(54-52)}$

$=\sqrt{54\times 39\times 13\times 2}$

$=\sqrt{9\times 6\times 13\times 3\times 13\times 2} = \sqrt{3^{2}\times 13^{2}\times 6^{2}} = 3\times 13\times 6=\boxed{234}$

We can use another form of Heron's formula:

$\frac{1}{4}\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}$

Which for our question can be written as:

$\frac{1}{4}\sqrt{4*15^2*41^2-(15^2+41^2-51^2)^2}$

This can be simplified to:

$\frac{1}{4}\sqrt{1512900-636804}$

Which can be further simplified to:

$\frac{1}{4}\sqrt{876096}$

which then equals to

$\frac{1}{4}*936$

So our answer is:

$\frac{936}{4}=234$

Answer:

$= \boxed{234}$

compute any angle by cosine law

$52^2 = 15^2 + 41^2 - 2(15)(41)cosB$

$B = 130.45 degrees$

$A$ $=$ $\frac{1}{2}$ $*15*41$ $*sin130.45$ $=234$

First find for the semi-perimeter:

s = ( 15 + 41 + 52 ) / 2 = 54

then using Heron's Formula:

A = sq. root of (54 x (54-15) x (54 - 41) x (54 - 52)) A = 234 sq. units :)

area of a scalene triangle (whose all sides are different) is given by (s(s-a)(s-b)(s-c))^0.5 where s=(a+b+c)/2, a,b,c being sides of triangle.

You can USE the Heron's Formula in solving the area of a triangle (if the lengths of 3 sides are already given)

Let a, b, c be the lengths of the sides of a triangle.

Area = square root of p(p-a) (p-b) (p-c)

where "p" is half the perimeter, or:

a + b + c / 2 [[this can be understand as, "the SUM of 3 sides DIVIDED by 2"]]

This is how it goes:

a = 15

b = 41

c = 52

p = 54 [[ 15+41+52 = 108 DIVIDED BY 2]]

A = square root of 54 (54 - 15) (54 - 41) (54 - 52)

A = square root of 54 (39) (13) (2)

A = square root of 54756

Therefore, the area is 234.. :)

The semi-perimeter of the triangle is (15+41+52)/2 whose answer is 54.Now by heron's formulae we can derive the area easily. The heron's formulae is the square root of {s(s-a)(s-b)(s-c)} where s is the semi-perimeter and a , b and c are the sides of the triangle.

sol:- let a =15, b=41,c=52 S=a+b+c/2 =15+41+52/2=>54 A=(s(s-a)(s-b)(s-c))^1/2 here a is area of triangle =(54(54-15)(54-41)(54-52))^1/2 =234

We know if 15^2 = 12^2 + 9^2 and 41^2 = 40^2 + 9^2 ,also 40 + 12 = 52. Because it can make a triangle has side 15, 41, 52. So, the areas is (9 x 52 )/2 = 9 x 26 = 234. Answer : 234