Recent questions in Angle Bisectors

Elementary geometryAnswered question

Kole Meyers 2023-03-11

The angles of a triangle are in the ratio 1 : 1 : 2. What is the largest angle in the triangle?

$A){90}^{\circ}\phantom{\rule{0ex}{0ex}}B){135}^{\circ}\phantom{\rule{0ex}{0ex}}C){45}^{\circ}\phantom{\rule{0ex}{0ex}}D){150}^{\circ}$

$A){90}^{\circ}\phantom{\rule{0ex}{0ex}}B){135}^{\circ}\phantom{\rule{0ex}{0ex}}C){45}^{\circ}\phantom{\rule{0ex}{0ex}}D){150}^{\circ}$

Elementary geometryAnswered question

Shayla Phelps 2023-03-05

How many right angles are required to create a complete angle?

1) Two

2) Three

3) Four

4) Five

1) Two

2) Three

3) Four

4) Five

Elementary geometryAnswered question

Bridger Joyce 2023-02-28

Find the sum of interior angles of a polygon with 12 sides in degrees.${1800}^{\circ}\phantom{\rule{0ex}{0ex}}{1200}^{\circ}\phantom{\rule{0ex}{0ex}}{1500}^{\circ}\phantom{\rule{0ex}{0ex}}{1400}^{\circ}$

Elementary geometryAnswered question

goldenlink7ydw 2023-02-17

How can I find an equation for the perpendicular bisector of the line segment that has the endpoints , (9,7) and (−3,−5)?

Elementary geometryAnswered question

ikalawangq00 2023-02-08

An angle formed by two opposite rays is called :

Zero angle

Complete angle

Right angle

Straight angle

Zero angle

Complete angle

Right angle

Straight angle

Elementary geometryAnswered question

Isaias Black 2023-01-22

Show that each diagonal of a rhombus bisects the angle through which it passes: ?

Elementary geometryAnswered question

animeagan0o8 2023-01-21

Find the central angle θ which forms a sector area 18 square feet of a circle of a radius of 10 feet?

Elementary geometryAnswered question

Isaias Black 2022-12-27

What value of y would make AOB, a line in the figure, if ∠AOC=4y and ∠BOC=(6y+30).

Elementary geometryAnswered question

vegetatzz8s 2022-11-25

Is it true that A ray is a bisector of an angle if and only if it splits the angle into two angles?

Elementary geometryAnswered question

Kareem Mejia 2022-11-17

Just like we have it in 2D coordinate geometry, is there an equation which describes the angle bisector of two straight lines in 3D coordinate geometry?

Elementary geometryAnswered question

vedentst9i 2022-11-14

Compute the coordinate equation of the angle bisectors of the planes E and F.

$E:x+4y+8z+50=0$ and $F:3x+4y+12z+82=0$

Proceed as follows:

a) Find the normal vectors of the two angle-bisecting planes.

b) Find a shablack point of planes E and F.

c) Now determine the equations of the two angle-bisecting planes.

I have the solutions but I don't understand why I must do things the way the solution is shown.

$\left|\left(\begin{array}{c}3\\ 4\\ 12\end{array}\right)\right|=13$

are the normal vectors from the equations. But this is not a good enough answer, all they asked for is the normal vectors, aren't these the normal vectors? Why must I add and subtract them like this?:

$13\cdot \left(\begin{array}{c}1\\ 4\\ 8\end{array}\right)+9\cdot \left(\begin{array}{c}3\\ 4\\ 12\end{array}\right)$

$13\cdot \left(\begin{array}{c}1\\ 4\\ 8\end{array}\right)-9\cdot \left(\begin{array}{c}3\\ 4\\ 12\end{array}\right)$

b) The solution says that I must choose one component, e.g: $x=2$ and then I substitute it into the equations and complete the simultaneous equation to find the point. Must it be only the x component? And why the value 2? Can it be any value? So according to the solution, the shablack point is $P(2,5,-9)$

c) The solution uses the answers from part a and b and gets this

$\left(\begin{array}{c}10\\ 22\\ 53\end{array}\right)\cdot [\left(\begin{array}{c}x\\ y\\ z\end{array}\right)-\left(\begin{array}{c}2\\ 5\\ -9\end{array}\right)]$

$\left(\begin{array}{c}-7\\ 8\\ -2\end{array}\right)\cdot [\left(\begin{array}{c}x\\ y\\ z\end{array}\right)-\left(\begin{array}{c}2\\ 5\\ -9\end{array}\right)]$

Is it a general rule to use the normal to find the equation from a shablack point?

$E:x+4y+8z+50=0$ and $F:3x+4y+12z+82=0$

Proceed as follows:

a) Find the normal vectors of the two angle-bisecting planes.

b) Find a shablack point of planes E and F.

c) Now determine the equations of the two angle-bisecting planes.

I have the solutions but I don't understand why I must do things the way the solution is shown.

$\left|\left(\begin{array}{c}3\\ 4\\ 12\end{array}\right)\right|=13$

are the normal vectors from the equations. But this is not a good enough answer, all they asked for is the normal vectors, aren't these the normal vectors? Why must I add and subtract them like this?:

$13\cdot \left(\begin{array}{c}1\\ 4\\ 8\end{array}\right)+9\cdot \left(\begin{array}{c}3\\ 4\\ 12\end{array}\right)$

$13\cdot \left(\begin{array}{c}1\\ 4\\ 8\end{array}\right)-9\cdot \left(\begin{array}{c}3\\ 4\\ 12\end{array}\right)$

b) The solution says that I must choose one component, e.g: $x=2$ and then I substitute it into the equations and complete the simultaneous equation to find the point. Must it be only the x component? And why the value 2? Can it be any value? So according to the solution, the shablack point is $P(2,5,-9)$

c) The solution uses the answers from part a and b and gets this

$\left(\begin{array}{c}10\\ 22\\ 53\end{array}\right)\cdot [\left(\begin{array}{c}x\\ y\\ z\end{array}\right)-\left(\begin{array}{c}2\\ 5\\ -9\end{array}\right)]$

$\left(\begin{array}{c}-7\\ 8\\ -2\end{array}\right)\cdot [\left(\begin{array}{c}x\\ y\\ z\end{array}\right)-\left(\begin{array}{c}2\\ 5\\ -9\end{array}\right)]$

Is it a general rule to use the normal to find the equation from a shablack point?

Elementary geometryAnswered question

Brenda Jordan 2022-11-08

The ratios of the lengths of the sides $BC$ and $AC$ of a triangle $ABC$ to the radius of the circumscribed circle are equal to 2 and $\frac{3}{2}$ respectively.Find the ratio of the lengths of the bisectors of internal angles of $B$ and $C$.

We are given $\frac{BC}{R}=\frac{a}{R}=2$ and $\frac{AC}{R}=\frac{b}{R}=\frac{3}{2}$,where $R$ is the circumradius of the triangle $ABC$.

$a=2R,b=\frac{3}{2}R$

I know that ${m}_{b}$=length of angle bisector of angle $B=\frac{2\sqrt{acs(s-b)}}{a+c}$ and ${m}_{c}$=length of angle bisector of angle $C=\frac{2\sqrt{abs(s-c)}}{a+b}$ but i need the third side in order to use these formulae,which i do not know.What should i do?

We are given $\frac{BC}{R}=\frac{a}{R}=2$ and $\frac{AC}{R}=\frac{b}{R}=\frac{3}{2}$,where $R$ is the circumradius of the triangle $ABC$.

$a=2R,b=\frac{3}{2}R$

I know that ${m}_{b}$=length of angle bisector of angle $B=\frac{2\sqrt{acs(s-b)}}{a+c}$ and ${m}_{c}$=length of angle bisector of angle $C=\frac{2\sqrt{abs(s-c)}}{a+b}$ but i need the third side in order to use these formulae,which i do not know.What should i do?

Elementary geometryAnswered question

Widersinnby7 2022-11-03

The angle bisector of

${L}_{1}:{a}_{1}x+{b}_{1}y+{c}_{1}=0$

and

${L}_{2}:{a}_{2}x+{b}_{2}y+{c}_{2}=0$

$({a}_{i},{b}_{i},{c}_{i})\in \mathbb{R}$

can be found be solving the equation $\frac{{a}_{1}x+{b}_{1}y+{c}_{1}}{\sqrt{{a}_{1}^{2}+{b}_{1}^{2}}}=\pm \text{}\frac{{a}_{2}x+{b}_{2}y+{c}_{2}}{\sqrt{{a}_{2}^{2}+{b}_{2}^{2}}}$

But our teacher told us that the equation of the angle bisector pasing through the region containing the origin can be obtained by solving only the positive case of the equation given that $({c}_{1},{c}_{2})>0$.How can we prove this?

${L}_{1}:{a}_{1}x+{b}_{1}y+{c}_{1}=0$

and

${L}_{2}:{a}_{2}x+{b}_{2}y+{c}_{2}=0$

$({a}_{i},{b}_{i},{c}_{i})\in \mathbb{R}$

can be found be solving the equation $\frac{{a}_{1}x+{b}_{1}y+{c}_{1}}{\sqrt{{a}_{1}^{2}+{b}_{1}^{2}}}=\pm \text{}\frac{{a}_{2}x+{b}_{2}y+{c}_{2}}{\sqrt{{a}_{2}^{2}+{b}_{2}^{2}}}$

But our teacher told us that the equation of the angle bisector pasing through the region containing the origin can be obtained by solving only the positive case of the equation given that $({c}_{1},{c}_{2})>0$.How can we prove this?

Elementary geometryAnswered question

Aldo Ashley 2022-10-22

The lengths of segments $PQ$ and $PR$ are 8 inches and 5 inches, respectively, and they make a 60-degree angle at P. Find the length of the angle bisector of angle R.

Through the Law of Sines I was able to find that the angle measure of R is approximately ${81.787}^{\circ}$ and from the Law of Cosines I figublack out that the measure of side $QR$ is 7 inches.

I am unsure of what steps I should take to find the measure of the angle bisector. Any help will be greatly appreciated!

Through the Law of Sines I was able to find that the angle measure of R is approximately ${81.787}^{\circ}$ and from the Law of Cosines I figublack out that the measure of side $QR$ is 7 inches.

I am unsure of what steps I should take to find the measure of the angle bisector. Any help will be greatly appreciated!

Elementary geometryAnswered question

snaketao0g 2022-10-20

How can I find a relation describing the length of angle bisector of regular polygon expressed as a function of its side's length?

For a equilateral triangle and a square with a side of length a, the relations are:

${t}_{a}=\frac{a\sqrt{3}}{2}$

and

${t}_{a}=\frac{a\sqrt{2}}{2}$

Could this be generalised to relation describing bisector's length of a regular N− polygon?

For a equilateral triangle and a square with a side of length a, the relations are:

${t}_{a}=\frac{a\sqrt{3}}{2}$

and

${t}_{a}=\frac{a\sqrt{2}}{2}$

Could this be generalised to relation describing bisector's length of a regular N− polygon?

Elementary geometryAnswered question

c0nman56 2022-10-20

I have seen in an old geometry textbook that the formula for the length of the angle bisector at $A$ in $\mathrm{\u25b3}\mathit{A}\mathit{B}\mathit{C}$ is

${m}_{a}=\sqrt{bc[1-{\left(\frac{a}{b+c}\right)}^{2}]},$

and I have seen in a much older geometry textbook that the formula for the length of the same angle bisector is

${m}_{a}=\frac{2}{b+c}\sqrt{bcs(s-a)}.$

(s denotes the semiperimeter of the triangle.)

I did not see such formulas in Euclid's Elements. Was either formula discoveblack by the ancient Greeks? May someone furnish a demonstration of either of them without using Stewart's Theorem and without using the Inscribed Angle Theorem?

${m}_{a}=\sqrt{bc[1-{\left(\frac{a}{b+c}\right)}^{2}]},$

and I have seen in a much older geometry textbook that the formula for the length of the same angle bisector is

${m}_{a}=\frac{2}{b+c}\sqrt{bcs(s-a)}.$

(s denotes the semiperimeter of the triangle.)

I did not see such formulas in Euclid's Elements. Was either formula discoveblack by the ancient Greeks? May someone furnish a demonstration of either of them without using Stewart's Theorem and without using the Inscribed Angle Theorem?

Elementary geometryAnswered question

Chaim Ferguson 2022-10-16

In triangle XYZ, the bisector of $\mathrm{\angle}XYZ$ intersects $\overline{XZ}$ at E if $\frac{XY}{YZ}=\frac{3}{4}$ an $XZ=42$, find the greatest integer value of XY.

Thus far, I have determined that $XE=18$ and $ZE=24$ by the angle bisector theorem, but I am unsure how to find XY.

Thus far, I have determined that $XE=18$ and $ZE=24$ by the angle bisector theorem, but I am unsure how to find XY.

Elementary geometryAnswered question

raapjeqp 2022-10-15

Prove that the external bisectors of the angles of a triangle meet the opposite sides in three collinear points.

I need to prove this using only Menelaus Theorem, Stewart's Theorem, Ceva's Theorem.

What I did:I tried by making a simple case diagram that is a diagram with obtuse angle in the given triangle. Then using Menelaus on angle bisectors with respect to the triangles and using angle bisector theorem for ratios of values.

I need to prove this using only Menelaus Theorem, Stewart's Theorem, Ceva's Theorem.

What I did:I tried by making a simple case diagram that is a diagram with obtuse angle in the given triangle. Then using Menelaus on angle bisectors with respect to the triangles and using angle bisector theorem for ratios of values.

Elementary geometryAnswered question

beefypy 2022-10-14

Prove that the sum of the reciprocals of the lengths of the interior angle bisectors of a triangle is greater than the sum of the reciprocals of the lengths of the sides of the triangle

I have tried different approaches to solve it here are some of them :

-The relation between the measures of angles in a triangle and the lengths of sides

-Trying triangle inequality on different triangles

-Using some formulas and inequalities involving angle bisectors and sides

Unfortunately, I wasn't able to solve it.

I don't want the full solution I just need some hints and suggestions on how to solve this problem.

I have tried different approaches to solve it here are some of them :

-The relation between the measures of angles in a triangle and the lengths of sides

-Trying triangle inequality on different triangles

-Using some formulas and inequalities involving angle bisectors and sides

Unfortunately, I wasn't able to solve it.

I don't want the full solution I just need some hints and suggestions on how to solve this problem.

Although it is called elementary geometry, dealing with the perpendicular angle bisectors is what you might remember from your school times. If you cannot remember the answers to those Geometry questions, it is best to take a closer look at the angle bisectors problems that we have collected based on the solutions that have been provided by our fellow students. The most important is to avoid checking only one example as it may have the graphs included without any verbal explanation as it may be required so! Now, others will have the simple solutions and more complex engineering tasks explained.