The legs of a right triangle are 5cm and 12 cm long. Find the lengths, to the nearest tenth, of the segments into which the bisector of the right angle divides the hypotenuse.
4.12 in
10.91 in
13 in
17 in
To find the lengths of the segments into which the bisector of the right angle divides the hypotenuse, we can apply the Angle Bisector Theorem.
1. Begin by drawing a right triangle. Label the legs as "a" and "b" (in this case, a = 5 cm and b = 12 cm) and the hypotenuse as "c".
/|
/ |
a / |
/ |
/____|
b
2. Calculate the length of the hypotenuse using the Pythagorean theorem:
c^2 = a^2 + b^2
c^2 = 5^2 + 12^2
c^2 = 25 + 144
c^2 = 169
c = √169
c = 13
Therefore, the length of the hypotenuse (c) is 13 cm.
3. Apply the Angle Bisector Theorem to find the lengths of the segments into which the bisector divides the hypotenuse. According to the theorem, the ratio of the two segments created by the bisector is equal to the ratio of the lengths of the two legs.
Let's assume that the bisector divides the hypotenuse into segments x and y.
x/y = a/b
Substitute the given values:
x/y = 5/12
Rearrange the equation to solve for x:
x = (5/12) * y
4. Substitute the value of x into the Pythagorean theorem to find the length of the segment y:
y^2 = a^2 + x^2
y^2 = 5^2 + [(5/12) * y]^2
Simplify the equation:
y^2 = 25 + (25/144) * y^2
(144/144) * y^2 = (3600/144) + (25/144) * y^2
(119/144) * y^2 = (3600/144)
Solve for y:
y^2 = (3600/144) * (144/119)
y^2 = 3600/119
y ≈ √30.2521
y ≈ 5.5 cm (rounded to the nearest tenth)
5. Substitute the value of y into the equation for x to find its length:
x = (5/12) * y
x = (5/12) * 5.5
x ≈ 2.2917 cm (rounded to the nearest tenth)
6. Therefore, the lengths of the segments into which the bisector of the right angle divides the hypotenuse are approximately 2.3 cm and 5.5 cm (rounded to the nearest tenth).