In a right triangle, if the median from the right angle to the hypotenuse is 5 in. what is the length of the hypotenuse?
The length of the median to the hypotenuse of a right triangle equals one-half the length of
the hypotenuse.
M = 1/2 hypotenuse
5 = 1/2 h
10 = h
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To find the length of the hypotenuse in 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 (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's label the three sides of the right triangle as follows:
- Hypotenuse: c
- One of the legs: a
- Median from the right angle to the hypotenuse: m
According to the problem, the length of the median is 5 in, which we can label as m = 5 in.
Since the median divides the hypotenuse into two equal segments, we can consider the two resulting right triangles. The two legs of these triangles are congruent, so we have a right isosceles triangle.
In an isosceles right triangle, the sides have a ratio of 1:1:√2.
Using this information, we can set up the equation:
a^2 + a^2 = c^2
2a^2 = c^2
Taking the square root of both sides:
√(2a^2) = √(c^2)
√2 * a = c
Hence, the length of the hypotenuse is √2 times the length of one of the legs.
In this case, since the median is the leg that is divided in half, we have m = a/2. Substituting this into the equation, we get:
c = √2 * (a/2)
c = √2 * a / √2
c = a / √2
Substituting the value of m = 5 in for a/2, we get:
c = 5 in / √2
To simplify this expression, we can rationalize the denominator by multiplying the numerator and denominator by √2:
c = (5 in / √2) * (√2 / √2)
c = (5 * √2) in
Therefore, the length of the hypotenuse is 5√2 inches.