In a 5 12 13 triangle what is the length of the bisector of the larger acute angle

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Table of Contents Show

Solution 1 (Angle Bisector Theorem)
Solution 2 (Analytic and Plane Geometry)
Solution 3 (Analytic and Plane Geometry)
Solution 4 (Distance Between a Point and a Line)
Solution 5 (Trigonometry)
Solution 6 (Trigonometry)
Solution 7 (Vectors)
Video Solution by TheBeautyofMath

The angle bisector of the acute angle formed at the origin by the graphs of the lines

and

has equation

What is

Diagram

~MRENTHUSIASM

Solution 1 (Angle Bisector Theorem)

This solution refers to the Diagram section.

Let

and

As shown below, note that

and

are on the lines

and

respectively. By the Distance Formula, we have

and

By the Angle Bisector Theorem, we get

Since

the answer is

Remark

The value of is known as the Golden Ratio:

~MRENTHUSIASM

Solution 2 (Analytic and Plane Geometry)

Consider the graphs of

and

. Since it will be easier to consider at unity, let

, then we have

and

Now, let

, and

. Cutting through side

of triangle

is the angle bisector

where

is on side

Hence, by the Angle Bisector Theorem, we get

By the Pythagorean Theorem,

and

. Therefore,

Since

, it is easy derive

The vertical distance between the -axis and

. Because the -coordinate of point

is , the slope we need to find is just the -coordinate

~Wilhelm Z

Solution 3 (Analytic and Plane Geometry)

Let's begin by drawing a triangle that starts at the origin. Assume that the base of the triangle goes to the point

. The line

is the hypotenuse of a right triangle with side length . The hypotenuse' length is

. Then, let's draw the line

. We extend it to when

. The length of the hypotenuse of the larger triangle is

with legs

. We then draw the angle bisector. We should label the triangle, so here we go.

is .

. When the line with angle

intersects the line

, call the point

. When the angle bisector intersects the line

, call the point

. By Angle Bisector Theorem,

. Since

is and

is , we have that

is . Solving for

, we get that

Since

, we have that

is just one more than that. Therefore,

. Since

is , we get that is

Remark

The answer turns out to be the golden ratio or phi (

). Phi has many properties and is related to the Fibonacci sequence. See Phi.

~Arcticturn

Solution 4 (Distance Between a Point and a Line)

Note that the distance between the point

to line

Because line

is a perpendicular bisector, a point on the line

must be equidistant from the two lines(

and

), call this point

Because, the line

passes through the origin, our requested value of

which is the slope of the angle bisector line, can be found when evaluating the value of

By the Distance from Point to Line formula we get the equation,

Note that

because

is higher than

and

because

is lower to

Thus, we solve the equation,

Thus, the value of

Thus, the answer is

(Fun Fact: The value

is the golden ratio

)

~NH14

Solution 5 (Trigonometry)

This problem can be trivialized using basic trig identities. Let the angle made by

and the -axis be

and the angle made by

and the -axis be

. Note that

and

, and this is why we named them as such. Let the angle made by

be denoted as

. Since

bisects the two lines, notice that

Now, we can take the tangent and apply the tangent subtraction formula:

Since

is increasing,

; thus,

~Indiiiigo

Solution 6 (Trigonometry)

Denote by

the acute angles formed between the -axis and lines

, respectively. Hence,

Denote by the acute angle formed by lines

and

Hence,

Following from the double-angle identity, we have

Hence,

Because is acute,

is acute. Hence,

. Hence,

Because line

is the angle bisector of , the angle between lines

and

Hence,

Therefore, the answer is

~Steven Chen (www.professorchenedu.com)

Solution 7 (Vectors)

When drawing the lines

and

, it is natural to choose points

and

together with origin

. See the figure attached. We utilize the fact that if

and are vectors of same length, then

bisects the angle between

and .

In particular, we scale the vector

by the factor of

to get

. So by adding vectors

and

we get

which bisects the acute angle formed by lines

and

. (In other words, quadrilateral

is a rhombus.) Finally, observe that

lies on the line

whose slope is

Thus, the answer is

~VensL.

Video Solution by TheBeautyofMath

https://youtu.be/ToiOlqWz3LY?t=504

~IceMatrix

In a 5 12 13 triangle what is the length of the bisector of the larger acute angle

Diagram

Solution 1 (Angle Bisector Theorem)

Solution 2 (Analytic and Plane Geometry)

Solution 3 (Analytic and Plane Geometry)

Solution 4 (Distance Between a Point and a Line)

Solution 5 (Trigonometry)

Solution 6 (Trigonometry)

Solution 7 (Vectors)

Video Solution by TheBeautyofMath

See Also

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