ABC is a right angled triangle . AD is bisector of angle BAC. Angle DAC = 20 degree . X = CD.

good luck if want the length of CD since we have no information on any of the side lengths.

However, we know from the angle bi sector theorem that BD/CD = AB/AC = cos 40

To solve this problem, we need to use the properties of a right-angled triangle and the angle bisector theorem. Here are the steps to find the value of X, which is CD:

Step 1: Draw the triangle ABC with AD bisecting angle BAC.

Step 2: Label the angle DAC as 20 degrees.

Step 3: Since AD is the angle bisector, it divides angle BAC into two equal angles. Therefore, angle BAD and angle CAD are equal. Thus, angle BAD = angle CAD = 20 degrees.

Step 4: In triangle ADC, we have angle CAD = 20 degrees and angle CDA = 90 degrees (since triangle ABC is right-angled). Therefore, the sum of angles CAD and CDA should be 180 degrees for triangle ADC.

Step 5: Using the angle sum property, we can calculate angle CAD by subtracting angles CAD and CDA from 180 degrees: angle CAD = 180 - 20 - 90 = 70 degrees.

Step 6: Now that we have the measure of angle CAD, we can apply the angle bisector theorem to find the length of CD. According to the angle bisector theorem, the ratio of the lengths of the sides of the triangle is equal to the ratio of the lengths of the other two sides. In this case, we can use the lengths of AD, AC, and CD:

AD / AC = CD / DC

Since AD is the angle bisector, we can assume AD = BD (since angle BAD and angle CAD are equal).

Step 7: Let's assume the length of CD as X. From step 6, we have:

BD / AC = X / (AC - X)

Since triangle ABC is right-angled, we can use the Pythagorean theorem to find the length of AC:

AC^2 = AB^2 + BC^2

Given that AB and BC are the other two sides of the right-angled triangle, substitute their values into the equation:

AC^2 = (AD + BD)^2 + BC^2

Since AD = BD, the equation becomes:

AC^2 = (2 × AD)^2 + BC^2

Step 8: Substitute the value of AC^2 from step 7 into the equation from step 6:

AD / sqrt((2 × AD)^2 + BC^2) = X / (sqrt((2 × AD)^2 + BC^2) - X)

Simplify this equation.

Step 9: Solve the equation for X. This will give us the length of CD.

AD = X × sqrt((2 × AD)^2 + BC^2) - X^2

AD × X = X × sqrt((2 × AD)^2 + BC^2) - X^2 × X

AD × X = X × sqrt((2 × AD)^2 + BC^2) - X^3

X^3 - AD × X + X × sqrt((2 × AD)^2 + BC^2) - AD × X = 0

X^3 - 2 × AD × X + X × sqrt((2 × AD)^2 + BC^2) = 0

Use a numeric method or approximation to solve the cubic equation for X.

To find the value of X, which represents the length of CD in the triangle ABC where AD is the bisector of angle BAC and angle DAC is 20 degrees, we can use the angle bisector theorem.

The angle bisector theorem states that in a triangle, if a line divides one of the angles in half, it divides the opposite side into segments that are proportional to the lengths of the adjacent sides.

In triangle ABC, let's consider the angle bisector AD. According to the angle bisector theorem, we can state the following relationship:

AB/BD = AC/CD

Since angle DAC is given as 20 degrees, we can apply this theorem to find the relationship between the sides of the triangle.

Let's assume AB = a, BD = b, and CD = X.
According to the known information, angle DAC is 20 degrees.

Applying the angle bisector theorem, we have:
a/b = AC/X

To solve for X, we need to find the ratio between the lengths of AC and CD. To do this, we need to find the length of AB, BD, and AC.

Next, we'll use the sine rule to find AB, BD, and AC:

AB/sin(BAC) = BC/sin(ABC)
a/sin(90 degrees) = BC/sin(B)
a = BC/sin(B)

BD/sin(BAC) = BC/sin(ACB)
b/sin(90 degrees) = BC/sin(C)
b = BC/sin(C)

Also, note that the sum of the angles in a triangle is always equal to 180 degrees:

B + 90 + C = 180
B + C = 90

Since one of the angles in a right-angled triangle is 90 degrees, the sum of the remaining two angles will also be 90 degrees.

Now, we have:
a = BC/sin(B)
b = BC/sin(C)
B + C = 90

We know that B + C = 90 degrees, so we can substitute that into our equations for a and b:

a = BC/sin(90 - a) = BC/cos(a)
b = BC/sin(90 - b) = BC/cos(b)

Now, let's simplify the equation using the angle bisector theorem:

a/b = AC/X

Substituting for a and b, we have:
(BC/cos(a))/(BC/cos(b)) = AC/X
cos(b)/cos(a) = AC/X

Since we know that angle DAC is 20 degrees, we can substitute b = 20 degrees and a = 90 - 20 = 70 degrees (since B + C = 90 degrees):

cos(20 degrees)/cos(70 degrees) = AC/X

Using a calculator, we can evaluate the cosine values and find the value of AC/X.

Finally, multiply AC/X by X to solve for X:
X = (AC/X) * X

By calculating the value of AC/X and simplifying the equation, you can find the value of X, which represents the length of CD in the given triangle.