the legs of a right angled triangle are 5 cm and 12cm long. find the lengths, to the tenth, of the segment into which the bisector of the right angle divides the hypotenuse.
Let ABC be the right-angled triangle with angleB = 90°
let the bisector angle B be BD where D is on the hypotenuse AC
We know that AD : DC = areaABD : areaBCD
areaABD = (1/2)(12)(BD)sin45°
areaBCD = (1/2)(5)(BD)sin45°
then areaABD : areaBCD = 12 : 5
so AD = (12/17)(13) = 9.2
and DC = 13-9.2 = 3.8
To find the lengths of the segments into which the bisector of the right angle divides the hypotenuse, we can use the concept of similar triangles. Let's start by drawing a right-angled triangle with legs that are 5 cm and 12 cm long.
First, let's label the triangle:
- The side opposite the right angle is the hypotenuse, which we'll call "c".
- The leg adjacent to one acute angle will be called "a", and the other leg will be called "b".
Let's calculate the length of the hypotenuse "c" 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 cm
Now, let's consider the line that bisects the right angle. This line divides the hypotenuse into two segments, let's call them "x" and "y". To find their lengths, we can use the concept of similar triangles.
The two smaller triangles formed by the bisector are both similar to the original triangle. By using the properties of similar triangles, we can set up the following proportions:
x / a = c / b
y / b = c / a
Let's substitute the values we know:
x / 5 = 13 / 12
y / 12 = 13 / 5
Now, we can cross-multiply and solve for the values of "x" and "y":
x = (13 / 12) * 5
x = 6.0833 cm (rounded to the nearest tenth)
y = (13 / 5) * 12
y = 31.2 cm (rounded to the nearest tenth)
Therefore, the lengths, rounded to the tenth, of the segments into which the bisector of the right angle divides the hypotenuse are approximately 6.1 cm and 31.2 cm.