Let's do this question in a right angle triangle, the length of the shortest median is 15 centimeter.
If the area of the triangle is two one six square centimeter, find the length of the longest median.
All right.
Was the video.
Give it a try and then let's do it together.
All right, we're back, I hope you have given it a try.
Let's do it together.
Let's draw a rough sketch.
We have a right angle triangle over here.
Now, you can observe that the shortest median will be to the longest side and the vice versa is also
true.
That is the longest median will be to the shortest side.
All right, now, this is a 90 degree angle right there for this side over here is the hypotenuse that
is it's the longest side.
Therefore, this median will be the shortest median right now.
It's given that the shortest median is equal to 15 centimetre.
Right.
Now, let me name this triangle as triangle ABC, now AC is equal to 30 centimeters, right?
Why is that so?
Because we have a right angle triangle and we have drawn a median to the hypotenuse, remember?
Therefore, if this is X, this also will be X and this also will be X.
If I call this point a SPE, then AP is equal to pieces equal to BP because point B is these Sock'Em
Center.
All right, so we have found that AC is equal to 30 centimeter centimetre right now, let me call abs
X and B C white.
So the length of ABS X and the length of B C is white.
Now, as per Pythagoras's theorem, we know that X squared plus Y square is equal to 30 square because
X the hypotenuse.
Right.
And it's also given that the area of the triangle is two hundred and sixteen now because this is the
right angle triangle, the altitude from point A is A itself.
We know that the area of a triangle is half base into height.
So the area of this triangle can be taken as 1/2 X and Y.
That's equal to two hundred and sixteen.
Right.
Therefore X and Y is equal to two hundred and sixteen in the two, which is equal to four hundred and
thirty two.
All right.
Now let's try to solve these two equations now before we go ahead and make it into a quadratic equation.
We can easily solve it using trial and error substitution.
Let's take an example.
We know that three, four and five are forming a Pythagorean Triplette.
Right.
And over here, we know one side is 30 centimetre.
That is the hypotenuse.
So if I want to get five to 30, I need to multiply it with six.
Right.
So five in the six gives me thirteen.
Now let me multiply three and four also with six.
So that gives me 18 and twenty four.
And let's check whether 18 and 24 satisfies X, Y is equal to 42.
Let's try that to 18 to twenty four.
If you do that you will get 42.
Therefore it is satisfied and hence X and Y is equal to 18 and 24.
All right.
Now we need to find the length of the longest median.
The length of the longest median will be to the shortest side.
So let's take Y as the shortest side.
So in that case, this will be the longest median right now.
Let's press on to this length over here.
That's BP is equal to nine centimetre because B is the point of B, C, right.
Because we are talking about a median eppy.
So we know B.C. has to be the shortest side.
And among eighteen twenty four, eighteen is the shortest one.
So therefore we have taken BK's 18 and we have got BP s nine.
All right.
Now let's proceed over here.
We know X is equal to twenty four.
Right.
So let's right that over here and we have a right angle.
Triangle A, B, B to aspart Pythagoras's theorem.
We know a B squared is equal to twenty four squared plus nine squared.
That gives us six hundred and fifty seven and eight.
B will be the square root of six hundred and fifty seven and that's equal to three root seventy three.
That is your answer.