The medians of a right triangle which are drawn from the vertices of the acute angle 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, which 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 other two sides.
Let's label the sides of the right triangle as follows:
- The medians drawn from the vertices of the acute angle will divide the triangle into three smaller triangles.
- Let's call the length of the median drawn to the hypotenuse "m".
- Let's call the lengths of the medians drawn to the other two sides "a" and "b".
According to the problem, we have:
- The length of the median drawn to one side is 5.
- The length of the median drawn to the other side is √40.
Using the properties of medians in a triangle, we know that the lengths of the medians divide each other into segments that have a 2:1 ratio. Therefore, we can write:
a = 2m
b = 2m
Now, we can use the Pythagorean Theorem. Let's consider the triangle with sides a, b, and the hypotenuse.
a² + b² = c²
Substituting the values of a and b we just found:
(2m)² + (2m)² = c²
4m² + 4m² = c²
8m² = c²
To find the length of the hypotenuse c, we need to find the value of m first. To do that, we can use the information given in the problem:
- The length of one median (a or b) is 5.
- The length of the other median (a or b) is √40.
Let's solve for m using the median length of 5:
4m² = 5²
4m² = 25
m² = 25/4
m² = 6.25
m = √6.25
m ≈ 2.5
So, the length of one of the medians is 2.5.
Now let's solve for m using the median length of √40:
4m² = (√40)²
4m² = 40
m² = 40/4
m² = 10
m = √10
Therefore, the length of the other median is √10.
Since there is only one hypotenuse in a right triangle, we can conclude that m = √10.
Now we can find the length of the hypotenuse c:
8m² = c²
8(√10)² = c²
8(10) = c²
80 = c²
c = √80
c = 4√5
Therefore, the length of the hypotenuse is 4√5.
To find the length of the hypotenuse of a right triangle, you 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 other two sides.
In this case, we are given the medians drawn from the vertices of the acute angle. Remember that a median is a line segment drawn from a vertex of a triangle to the midpoint of the opposite side.
Let's assume that the acute angle of the right triangle is angle A, and the medians drawn from the vertices of angle A are represented as AM, AN, and AP. Let's also assume that AM = AN = 5 and AP = √40.
First, we need to find the lengths of the other two sides, which are AM and AN.
Since AM and AN are medians, they divide the opposite sides in a 2:1 ratio. Let's denote the length of side BC, opposite to angle A, as 2x. Then, we can represent the length of AM as x, and the length of AN as x.
Now, we have a right triangle with the following side lengths:
AM = x = 5
AN = x = 5
BC = 2x
According to the Pythagorean theorem, the square of the hypotenuse (BC) is equal to the sum of the squares of the other two sides (AM and AN):
BC^2 = AM^2 + AN^2
(2x)^2 = x^2 + x^2
4x^2 = 2x^2
2x^2 = 40
Dividing both sides by 2, we get:
x^2 = 20
Taking the square root of both sides, we find:
x = √20
Now that we know x, we can find the length of the hypotenuse BC:
BC = 2x = 2 * √20 = 2√(4 * 5) = 2 * 2√5 = 4√5
Therefore, the length of the hypotenuse BC is 4√5.