The length of a hypotenuse of a right triangle is 29 feet. The Altitude drawn perpendicular to the hypotenuse is 10 feet. What are the lengths of the two legs?
by similar triangles
divide hypotenuse into x and 29-x
x/10 = 10/(29-x)
100 = 29 x - x^2
x^2 -29 x + 100 = 0
(x-25)(x-4) = 0
so the little triangle has short arm 4 and the big one has long arm 25
sqrt (100 + 16) = sqrt(116) = 2 sqrt(29)
sqrt (100+625) = sqrt(725) = 5 sqrt(29)
check
4(29) + 25(29) = 29^2 yes
To find the lengths of the two legs of a right triangle, we can make use of the Pythagorean theorem, which 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 (the legs).
Let's denote the length of one leg as 'a' and the other leg as 'b'. We know that the hypotenuse (c) is 29 feet and the altitude (h) is 10 feet.
Using the Pythagorean theorem, we have the equation:
a^2 + b^2 = c^2
Substituting the given values, we get:
a^2 + b^2 = 29^2
Now, we also know that the altitude divides the right triangle into two smaller triangles that are similar to the original triangle.
By applying the property of similar triangles, we can determine that the altitude divides the hypotenuse into two segments, such that the lengths of the segments are proportional to the lengths of the corresponding legs.
Let's denote the lengths of the segments of the hypotenuse as 'x' and 'y'. Therefore, we have:
x + y = c
Since the altitude is perpendicular to the hypotenuse, the two segments of the hypotenuse, x and y, along with the altitude h, form two right triangles. Using the Pythagorean theorem for both triangles, we have the following equations:
x^2 + h^2 = a^2
y^2 + h^2 = b^2
Substituting the given values, we have:
x^2 + 10^2 = a^2
y^2 + 10^2 = b^2
Now, we have a system of equations:
a^2 + b^2 = 29^2
x^2 + 10^2 = a^2
y^2 + 10^2 = b^2
x + y = 29
We can now solve this system of equations to find the lengths of the two legs, a and b.