How to find the third side of an isosceles triangle with only 2 sides known

Q: How do you find the third side of a triangle given two sides?

(Perpendicular)2 + (Base)2 = (Hypotenuse)2 Using the above equation third side can be calculated if two sides are known.

Question

Video transcript

- [Instructor] We're asked to find the value of x in the isosceles triangle shown below. So that is the base of this triangle. So pause this video and see if you can figure that out. Well the key realization to solve this is to realize that this altitude that they dropped, this is going to form a right angle here and a right angle here and notice, both of these triangles, because this whole thing is an isosceles triangle, we're going to have two angles that are the same. This angle, is the same as that angle. Because it's an isosceles triangle, this 90 degrees is the same as that 90 degrees. And so the third angle needs to be the same. So that is going to be the same as that right over there. And since you have two angles that are the same and you have a side between them that is the same this altitude of 12 is on both triangles, we know that both of these triangles are congruent. So they're both going to have 13 they're going to have one side that's 13, one side that is 12 and so this and this side are going to be the same. So this is going to be x over two and this is going to be x over two. And so now we can use that information and the fact and the Pythagorean Theorem to solve for x. Let's use the Pythagorean Theorem on this right triangle on the right hand side. We can say that x over two squared that's the base right over here this side right over here. We can write that x over two squared plus the other side plus 12 squared is going to be equal to our hypotenuse squared. Is going to be equal to 13 squared. This is just the Pythagorean Theorem now. And so we can simplify. This is going to be x. We'll give that the same color. This is going to be x squared over four. That's just x squared over two squared plus 144 144 is equal to 13 squared is 169. Now I can subtract 144 from both sides. I'm gonna try to solve for x. That's the whole goal here. So subtracting 144 from both sides and what do we get? On the left hand side, we have x squared over four is equal to 169 minus 144. Let's see, 69 minus 44 is 25. So this is going to be equal to 25. We can multiply both sides by four to isolate the x squared. And so we get x squared is equal to 25 times four is equal to 100. Now, if you're just looking this purely mathematically and say, x could be positive or negative 10. But since we're dealing with distances, we know that we want the positive value of it. So x is equal to the principle root of 100 which is equal to positive 10. So there you have it. We have solved for x. This distance right here, the whole thing, the whole thing is going to be equal to 10. Half of that is going to be five. So if we just looked at this length over here. I'm doing that in the same column, let me see. So this length right over here, that's going to be five and indeed, five squared plus 12 squared, that's 25 plus 144 is 169, 13 squared. So the key of realization here is isosceles triangle, the altitudes splits it into two congruent right triangles and so it also splits this base into two. So this is x over two and this is x over two. And we use that information and the Pythagorean Theorem to solve for x.

Answer

Verified

Hint: Problems of this type have non-specific answers, this means that we will be able to find the range between which the answer lies. Using a trigonometric formula of isosceles triangle, we will get the limits between which the length of the third side of an isosceles triangle can exist. So, we can take any value between the limits and conclude it as the answer to the problem.

Complete step by step answer:

For an isosceles triangle if the given two sides have the same length then for calculating the length of the third side of the triangle, we can use a trigonometric formula for finding the length of the unknown side of the triangle, which is
$l=2\cdot a\cdot sin\left( \dfrac{\theta }{2} \right)$
Here, $l$ is the length of the third side of the triangle, $a$ is the length of the other two sides of the triangle and $\theta $ is the angle between the similar sides of the triangle.
The angle $\theta $ lies between $0$ to $\pi $
We know that $\sin \left( \dfrac{0}{2} \right)=0$ and $\sin \left( \dfrac{\pi }{2} \right)=1$
From the formula we get
$l=2\cdot 15\cdot sin\left( \dfrac{0}{2} \right)$
$\Rightarrow l=0$ , when $\theta =0$
Also, $l=2\cdot 15\cdot sin\left( \dfrac{\pi }{2} \right)$
$\Rightarrow l=30$ , when $\theta =\pi $
As the angle $\theta $ can take any value between the range $\left( 0,\pi \right)$ the length of the third side of an isosceles triangle can take any value between the range $\left( 0,30 \right)$ .
Therefore, we can conclude that the third side of an isosceles triangle can be of any length between $0$ and $30$ .

Note:
The problem can also be solved by applying the property of triangles. According to a property of triangles the sum of any two sides is greater than the third and the difference between any two sides is less than the third. So, in this case the lower limit for the length of the third side is $0$ and the upper limit for the length of the third side is $30$ . As, $15-15=0$ and $15+15=30$ .