The medians of the legs of a right triangle are 3 square root 6 and 6. What is the length of the hypotenuse?
Here is the same problem with different numbers.
The medians of a right triangle which are drawn from the vertices of the acute angles are 5 and root 40. What is the length of the hypotenuse? Can someone show me how to do it? thanks
Draw triangle ABC where A is the upper left vertex, B is the lower right vertex and C is the right angle at the lower left. Call AC x and BC y. Draw the medians BD and AE, D being the midpoint of x and E the midpoint of y.
From triangle ACE, x^2 + (y/2)^2 = 25 or 4x^2 + y^2 -100 = 0.---(1)
From triangle DCB, (x/2)^2 + y^2 = 40 or x^2 + 4y^2 - 160 = 0.---(2). (2)
From (1), y^2 = 100 - 4x^2
Substituting into (2) yields x = 4 and y = 6.
Therefore, the hypotenuse is sqrt(4^2 + 6^2 = 7.211.
To find the length of the hypotenuse of a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.
In this case, let's assume that the lengths of the two legs are a and b, and the length of the hypotenuse is c. According to the problem, the medians of the legs are given as 3√6 and 6.
Now, the medians of a triangle divide each side in the ratio 2:1. This means that the length of one leg, a, is twice the given median, 3√6, and the length of the other leg, b, is twice the given median, 6.
So, we have a = 2 * 3√6 = 6√6 and b = 2 * 6 = 12.
Now, we can use the Pythagorean theorem:
c^2 = a^2 + b^2
Substituting the values we found, we have:
c^2 = (6√6)^2 + 12^2
c^2 = 36*6 + 144
c^2 = 216 + 144
c^2 = 360
Taking the square root of both sides, we find:
c = √360
Simplifying the square root, we have:
c = 6√10
Therefore, the length of the hypotenuse is 6√10.