How to find the length of third side of a triangle given two sides & area of the triangle.?

Question

How to find the length of third side of a triangle given two sides & area of the triangle ?
Please tell the answer which will be a solution for a high school standard.

How to find the length of third side of a triangle given two sides & area of the triangle ?
Please tell the answer which will be a solution for a high school standard.

Adding some details :-
1. No mention that, it is a right angled triangle.
2. Since this is a problem of a high school student, solution involving SIN/COS functions does not suit properly.
3. We may use the basic formulae like
A = 1/2.b.h
or, A = sqrt(s(s-a)(s-b)(s-c))
But, issues is it will give an equation of power 4, which will be difficult for a high school student :-(

Answer 1

If two sides are given and you need to know the length of the third side use the Pythagrean Theorem formula

c² = a² + b²

- - - - - - - - -s-

Answer 2

As you say in your notes, deriving the sine or cosine is not easy here, so the formulas using trignometric functions are not the answer.

I'm thinking that formula A = √(s(s-a)(s-b)(s-c)) does the trick. Yes, it produces a degree four polynomial, but the polynomial is of the form ax^4 + bx² + c, as is shown below. Given that, you can reduce it to a quadratic by the substitution z = x², turning the polynomial into the form az² + bz + c.

As for resolving the formula, assume a and b are the known sides, and c is the unknown side. Then:

A = √(s(s-a)(s-b)(s-c))
A² = s(s-a)(s-b)(s-c)

Since s = (a + b + c)/2:

A² = ((a + b + c)(a + b - c)(a - b + c)(b - a + c)/16
16A² = ((a + b + c)(a + b - c) * (a - b + c)(b - a + c)

Now, (a + b + c)(a + b - c) can be factored as (a + b)² - c², and (a - b + c)(b - a + c) can be factored as c² - (a-b)². So, we can write the last equation as

16A² = ((a + b)² - c²)(c² - (a - b)²)

...and, while the final equation looks messy enough, it is clear that, when a, b, and A are known, you can reduce the equation to a quadratic by the substitution z = c². That gives you a way to solve for c. (If you think about it, solving for c won't be all that messy when you plug in values for a, b, and A.)

Answer 3

In the "Additional Details" asker nearly answers the question himself, so I assume this is being presented as a puzzle rather than a question as such.

Some of the first answers assumed a right triangle and mentioned the Pythagorean Theorem, but as mentioned in the "Additional Details: it is not stated that it is a right triangle. But then the asker points out how to divide the triangle into two right triangles. A previous answer suggested bisecting the base, but it is not given that it is an Isosceles triangle so that doesn't work. However if one determines the height, using the formula given by the asker, (A = 1/2.b.h) then one can derive two right triangles and solve with the Pythagorean Theorem.

Answer 4

Let the base of the triangleabc be AB=a , the other two sides are BC=b and CA=c.
We know the values of 2 sides . For simplification I assume a= 8units , and c= 5 units. b is not known to us.

So Draw a perpendicular to the base AB which divides AB in to two equal parts at x.So XA=XB(4 unites each).
Now the triangle XAC,is right angled triangle which means cA^= Ax^2+xC^2 => 5^2=4^2 + xC^2

So xC= 3. From this we can find the side BC, the triangle xBC is right angled triangle , So
BC^2=xB^2+Cx^2
BC^2= 4^2+3^2 , so BC=3 units. So AB=8 , BC=3, and CA=5. This doesn,t even need the area too to find the third side.

Answer 5

If the two sides of a triangle are at 90 degrees and both the lines are 1000 mm long. How to apply formula to find the length of the third line.

Answer 6

This Site Might Help You.

RE:
How to find the length of third side of a triangle given two sides & area of the triangle.?
How to find the length of third side of a triangle given two sides & area of the triangle ?
Please tell the answer which will be a solution for a high school standard.

Answer 7

(1) Let A,B, be the known sides, and let w be the angle between them. From elementary trigonometry (use of the law of sines) we have that the area S of the triangle is given by:
S = (AB)sin(w)/2 ==> sin(w) = 2S/AB, and from here you get the value of the angle w (just calculate the inverse function of sin, arcsin, of 2S/AB. You can do this in some calculators with the keys SHIFT and then SIN, or INV and then SIN).
Let X be the unknown length of the third side, and use now the law of cosines:
x^2 = A^2 + B^2 - 2ABcos(w)
And since you already know w and thus cos(w), you'll get x^2, and taking square roots, also x.
Regards
Tonio

Answer 8

Please make a sketch of triangle with given data's; two sides (a and b) and area (A) of the triangle! Further 'c' is the third side we intend to work out a length!

Naming 'a' as longer side among 'a' and 'b'...

Consider that sides 'a', and 'b' has a merging-vertex (C)!

Now "draw a perpendicular of side a" through "end point of line b", (which is an opposite corner of 'a'). Said perpendicular cuts line 'a' at point 'D'

Now we have two right angled triangles that are similar triangles! Further, resulting two areas maintain an area relation A1 : A2 (A1 is right angled triangle area touching on side "b" and A2 is other right angled triangle).

Said two similar triangles maintain an easy to apply area-length relation!

One of these relations A1/A2 = b / c, (in which 'c' is unknown third side) enable us to solve the problem completely! Reason is 'b' and 'c' are hypotanuses of respective similar right angled triangles and AD is common side of both right angled triangles!

Problem solving is thus fixed which a high scool student can do without any great effort!

Regards!

Answer 9

Let us assume the known sides to be "a" & "b", the unknown side being "c".

Let us assume, the hieght of the triangle perpendicular to side "a" be "h".

Area of the triangle, A = (1/2)*a*h
=> h = (2*A)/a ------- (1)

Let us assume that the perpendicular drawn to side "a" dvides the side "a" into "a1" & "a2".
i.e., a1+a2 = a --------- (2)

Then we have the following equations (for right angled triangles)

b2 = h2 + a12 ---- (3)
c2 = h2 + a22 ---- (4)

From Eq(3), a12 = b2-h2
=> a1 = Sqrt(b2-h2) ---- (5)

From Eq(2), a2 = a-a1
=> a2 = a - sqrt(b2-h2) ----- (6)

From Eq(4) & Eq(6)
c2 = h2 + (a-sqrt(b2-h2))2

=> c = sqrt(h2+(a-sqrt(b2-h2))2)

In this way we can find out the Third length of the triangle.

Answer 10

if its a right triangle than u use a2+b2=c2 (the 2 is supposed to be a squared...sry) if not than u can use this formula... a=1/2LH just plug in the area and the lenght or the hight(wich ever one you kno) and solve from there

Answer 11

first of all apply heros formula and take out a equation in which area and the remaining sides will be there and from there find it


or drop a alltitude and find its length and from that take out the side length