Angle Bisector
- What is an Angle Bisector?
- How to Construct an Angle Bisector?
- Angle Bisector in a Triangle
- Angle Bisector Theorem
- Properties of Angle Bisector
- Real-Life Applications of Angle Bisectors
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What is an Angle Bisector?
An angle bisector is defined as a line, a ray, or a line segment that divides an angle into two equal parts or two angles of equal measure. After dividing the angle by an angle bisector the angle measures of the two new angles are the same. In geometry, an angle bisector helps in dividing an angle into perfectly equal parts.
The following image shows an angle bisector BD that divides the angle `∠ABC` into two angles `∠ABD` and `∠DBC`.
Note that `∠ABD= ∠DBC`
How to Construct an Angle Bisector?
An angle bisector is easy to construct with geometrical instruments, a compass, and a straightedge ruler. Consider the image `∠PQR` shown above. To construct an angle bisector following steps can be followed.
Step `1`: Draw the `∠PQR` with ruler and compass fixing `Q` as vertex and `P` and `R` as two vertices running out from `Q`.
Step `2`: Now, place the compass at point `Q`. Then, open the compass with a length greater than half the length of the ray `QP`.
Step `3`: Mark arcs with the opened compass with `Q` as a fixed point on line `QP` and `QR` respectively.
Step `4`: Mark, the intersection of arcs and lines `QR` and `QP` with points `S` and `T` respectively.
Step `5`: Now, put the pointer on point `S` and draw an arc with the same radius. Similarly, put the pointer on `T` and draw another arc intersecting the previous arc. Mark the intersection point of arcs as `U`.
Step `6`: Join `Q` and `U` and extend through `U`. Line `QU` is the angular bisector of `∠PQR`.
Angle Bisector in a triangle
Any triangle has three angle bisectors. A triangle has three vertices and three sides. So, these bisectors originate from the vertices of the triangle and fall on the sides opposite to the vertices of the triangle. The bisectors of a triangle divide the angles of the triangle into two angles of equal measure.
A triangle `ABC` is shown below. The angle bisector from vertex `A` will fall on side `BC`. Similarly, the angle bisector from vertex `B` will fall on side `AC`, and the angle bisector from vertex `C` falls on side `AB`.
In the image above, `AD`, `BF`, and `CE` are angle bisectors of `∠BAC , ∠ABC and ∠BCA`. The angle bisectors intersect at point `G`. So, point `G` is called the ‘incenter’ of the triangle.
One important observation in the image is that point `G` is equidistant from all the sides of the triangle.
Angle Bisector Theorem
The angle bisector theorem states that “In a triangle when an angle bisector is drawn from one vertex and it falls on the opposite side of the triangle, then the ratio of the lengths of the two segments into which the opposite has been divided into is equal to the ratio of the other two sides.”
Mathematically, it can be expressed by the above image. In `GHI`, `IJ` is the angle bisector of `∠GIH`. According to the angle bisector theorem.
`c/d=a/b`
Properties of Angle Bisector
The following properties of an angle bisector can be understood:
- An angle bisector is a ray or line that divides the angle into two equal parts.
- In a triangle, the angle bisectors are at equal distances from the sides of the triangle.
- The incenter of the triangle is the center of the circle of maximum radius that can be inscribed in a triangle.
- The angle bisector divides the opposite sides of the triangle into a ratio that is equal to the ratio of the other two sides.
Real-Life Applications of Angle Bisectors
At noon when the sun is above an upright object, the angle bisector of the angle between the subject and the shadow is used to estimate the sun's position in the sky, this technique is very useful in the positioning of the solar panels.
In civil engineering, the building engineer uses the angle bisector to ensure that the corners of the wall meet at right angles.
In agriculture, angle bisectors are used by the farmers to judge for spacing between the seeds sown for maximum sunlight exposure to crop plants.
In the image below, `BD` is the angle bisector of the `∠ABC` which is `90°`, which divides it into two equal angles of measure `45°`. Therefore,
`∠ABD= ∠DBC=45°`
Solved Examples
Example `1`: Calculate `∠ABD`, if `BD` is the angle bisector of `∠ABC` and `∠ABC=80°`.
Solution:
Since `BD` is the angle bisector of `∠ABC`, It will divide `∠ABC` into two angles of equal measure.
Therefore, `∠ABD=(∠ABC)/2=(80°)/2=40°`.
Example `2`: In the given figure, ray `BC` bisects `∠ABD`, and `∠ABC=44°`. Calculate `∠CBE`.
Solution:
Here, `∠ABC=44°`.
Since `BC` is the bisector of the `∠ABD`. Therefore, `∠CBD=44°`.
Then, `∠CBE=∠CBD-∠EBD=44°-29°=15°`.
`∠CBE=15°`
Example `3`: In the image below, find the value of `x`, if ray `OP` is the angle bisector of `∠ROS`.
Solution:
Here, `OP` is the angle bisector of `∠ROS`.
Therefore, `∠ROP=∠POS`
Then,
`5x+4=34`
`5x=34-4`
`5x=30`
`x=6`
Example `4`: Traingle `ABC` has a side `AB` of measure `8` cm. If the angle bisector from vertex `A` divides the opposite side `BC` into measures of `4` cm and `6` cm. Find the measure of the third side.
Solution:
Here, `AB = 8` cm and the side `BC = 10` cm, which is divided by the bisector as `BD = 4` cm and `DC = 6` cm.
Now, applying the angle bisector theorem, we can write the ratios as
`(AB)/(AC)=(BD)/(DC)`
Put the values.
`8/(AC)=4/6`
Cross multiply.
`8/4=(AC)/6`
`AC=(8\times 6)/4`
`AC=12` cm
The measure of the third side of the triangle `ABC` is `12` cm.
Practice Problems
Q`1`. In the figure below, `BF` is the angle bisector of `∠ABC` and `BE` is the angle bisector of `∠CBD`. `∠ABF=14°` and `∠EBD=19°`. Find `∠ABD`.
- `76°`
- `66°`
- `56°`
- `46°`
Answer: b
Q`2`. `OR` is the angle bisector of `∠QOP`. Find `∠QOS`.
- `60°`
- `76°`
- `58°`
- `68°`
Answer: c
Q`3`. In a `XYZ`, if `XP` is the bisector of `∠X`. `XP` divides line `YZ` into two parts with length `YP = 5` cm and `PZ = 6` cm. If the length of `XY = 15` cm. Find the length of `XZ`.
- `18` cm
- `20` cm
- `25` cm
- `10` cm
Answer: a
Q`4`. If `AB = AC` in the triangle shown below, and `AD` is the bisector of `∠A`, which of the following statements is true?
- `BC = BD`
- `AD = BC`
- `AC = BD`
- `BD = CD`
Answer: d
Frequently Asked Questions
Q`1`. Can an angle bisector divide the triangle into two equal parts?
Answer: No, angle bisectors do not necessarily divide the triangles into two equal parts. However, they divide an angle into angles of the same measure.
Q`2`. Are the angle bisectors and perpendicular bisectors the same?
Answer: No, they are completely different. An angle bisector divides the angle into two equal parts, whereas the perpendicular bisector divides a line segment into two equal parts forming right angles with the line segment.
Q`3`. Do the angle bisectors divide the opposite sides into two equal parts?
Answer: Angle bisectors can divide the opposite sides into two equal parts with the condition that the triangle must be equilateral or isosceles. In the case of the isosceles triangle, the bisector must be falling on the unequal side.
