The sides of a triangle are 15, 20 and 28. How long are the segments into which the bisector of the largest angle separates the opposite side
The largest angle will be opposite the side 28
let it be 2Ø, so each bisected angle is Ø
let the bisector form angles A and B along the 28 side, so that A + B = 180° --> B = 180-A
and we know sinA = sin(180-A) = sinB
let the 28 side be split into x and 28-x, where x is adjacent the side 20
Now use the sine law in each of the smaller triangles
sinØ/x = sinA/20
sinØ = x sinA/20
sinØ/(28-x) = sinB/15
sinØ = (28-x)sinB/15
thus:
x sinA/20 = (28-x)sinB/15 , but remember sinA = sinB, so dividing them out
x/20 = (28-x)/15
15x = 560 - 20x
35x = 560
x = 16
so the side 28 is cut into parts 16 and 12
they are in the ratio 3:4 as provided by the angle bisector theorem.
good call Steve!
(at least I got the right answer, lol)
Ah, triangles, the lovechild of shapes. Now, let's see how we can slice and dice this one. The bisector of the largest angle, huh? Well, here's a fun fact: when you have a triangle, the angle bisector divides the opposite side into segments that are proportional to the other two sides.
So, let's break it down. We have a triangle with sides 15, 20, and 28. Now, we know the largest angle is opposite the longest side, which in this case is 28. So, when we draw the bisector of that angle, it will split the opposite side into two segments, let's call them "x" and "y".
Since the bisector divides the opposite side proportionally, we can set up a handy-dandy little equation. We know that x/y is equal to the lengths of the other two sides, which are 15 and 20.
So, x/y = 15/20. Now, let's simplify that. Dividing both sides by 5, we get x/y = 3/4.
Now, we can solve for x or y. Let's pick x. Multiplying both sides of the equation by y, we have xy = 3y/4.
And there you have it. The segments into which the bisector separates the opposite side are x = 3y/4 and y = 4x/3. Now, all we need are some numbers to plug in, and voilà!
To find the lengths of the segments into which the bisector of the largest angle separates the opposite side of a triangle, we can use the angle bisector theorem.
The angle bisector theorem states that if a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the lengths of the other two sides of the triangle.
In this case, the largest angle of the triangle is opposite the side of length 28. Let's denote the lengths of the segments created by the angle bisector as x and y.
According to the angle bisector theorem, we have the following proportion:
x/y = 15/20
To find the values of x and y, we can cross-multiply:
x * 20 = y * 15
Simplifying this equation, we have:
20x = 15y
Now, we can substitute the given side lengths into the equation:
20x = 15y
20x = 15(28 - x)
Simplifying further:
20x = 420 - 15x
Adding 15x to both sides:
35x = 420
Dividing both sides by 35:
x = 12
Now, we can find the value of y by substituting the value of x back into the equation:
y = (15/20) * x
y = (15/20) * 12
y = 9
Therefore, the lengths of the segments into which the bisector of the largest angle separates the opposite side are 12 and 9.