One side of triangle is 10 centimetres, two angles next to that side are 30° and 60°. I need to find all three angle bisectors of triangle.
You clearly have a 30-60-90 right-angled triangle where the hypotenuse is 10
So the sides are 5 , 5sqrt(3) and 10.
Did you want to find the lengths of the angle bisectors.
Simple trig calculations would do that.
Give it a shot.
It does not. Or I don‘t know what you‘re talking about.
Is these right if you can check, I tried to do it on my own.
10/3√3 ; 10√3/√2+√6 .
didnt get the last one...
Make a sketch, the 30-60-90 triangle has side ratios of 1 : √3 : 2.
Yours is similar to that by a factor of 5
You will have a triangle where the side opposite the 30° angle is 5, the side opposite the 60° angle is 5√3
and the hypotenuse is 10
(check: 5^2 + (5√3)^2 = 10^2 ?, yup!)
Now do one angle bisector at a time:
the 60° bisector, let it be of length x
cos 30° = x/5
x = 5cos 30 = 5(√3/2) = 5√3/2 or appr. 4.33
the 30° bisector, let its length be y
cos15° = y/(5√3)
y = 5√3 cos15 = appr. 8.37
the 90° bisector, let it be z
I will use the Sine Law
z/sin60 = 5/sin75 , (use angle sum of triangle to get 75)
z = 5sin60/sin75 = appr. 4.48
To find the angle bisectors of a triangle, you need to determine the measure of each angle in the triangle. In this case, we are given one side of the triangle, along with the measures of two angles adjacent to that side. We can use this information to find the measure of the third angle and then proceed to find the angle bisectors.
Let's start by finding the measure of the third angle. We know that the sum of all angles in a triangle is always 180°. So, we can use this information to find the missing angle.
Given:
One side of the triangle: 10 centimeters
Adjacent angles: 30° and 60°
Step 1: Find the measure of the third angle.
Sum of all angles in a triangle = 180°
Let the third angle be 'x'.
30° + 60° + x = 180°
Combining like terms:
90° + x = 180°
Subtracting 90° from both sides:
x = 180° - 90°
x = 90°
Therefore, the measure of the third angle is 90°.
Now that we know the measures of all three angles in the triangle (30°, 60°, and 90°), we can proceed to find the angle bisectors.
Step 2: Find the angle bisectors.
An angle bisector of a triangle divides an angle into two equal parts. To find the angle bisectors, we need to locate the incenter of the triangle, which is the point where all three angle bisectors intersect.
To find the incenter, we need to draw the perpendicular bisectors of the triangle's sides. The perpendicular bisectors will intersect at the incenter.
Let's label the vertices of the triangle as A, B, and C, where the side of length 10 centimeters is opposite vertex C.
1. Draw the line segment AB with length 10 centimeters.
2. At points A and B, draw arcs of the same radius to intersect at point D.
3. Draw the perpendicular bisector of AB. It will pass through point D and intersect AB at point E.
4. Repeat steps 2 and 3 for line segments AC and BC to find the perpendicular bisectors and their intersections with the respective sides.
5. The point where all three perpendicular bisectors intersect is the incenter of the triangle.
6. From the incenter, draw the three angle bisectors to the opposite sides of the triangle. These angle bisectors will divide the angles into two equal parts.
Note: Drawing the construction accurately can be done using a ruler, compass, and protractor.
By following these steps and constructing the perpendicular bisectors, you will be able to identify the incenter and draw the angle bisectors of the triangle.