Trigonometry query. ABC is a rightangled triangle. AD is the bisector of angle BAC. Angle DAC = 15 degrees. X = CD. find X. I know the answer is 7.1 but cannot work out the theory. Could you please help?
Thankyou.
To find the value of X, we can apply the angle bisector theorem in triangle ABC. According to the theorem, the angle bisector of a triangle divides the opposite side into segments that are proportional to the lengths of the adjacent sides.
In our case, AD is the angle bisector of angle BAC. Let's label the length of AB as a, BC as b, and AC as c.
Since AD is an angle bisector, it divides side BC into two segments, BD and CD. We are given that angle DAC is 15 degrees.
Now, according to the angle bisector theorem, we have the following proportion:
BD/CD = AB/AC ---(1)
Given:
angle DAC = 15 degrees
We know that angle BAC is a right angle (90 degrees).
Therefore, angle BAD is equal to 90 - 15 = 75 degrees.
In triangle BAD, we can apply the trigonometric function tangent to find the length of BD.
tan(75) = BD/AD
Now, let's find the value of BD. We can rearrange the equation as:
BD = tan(75) * AD
Next, let's work on finding the value of AD.
In triangle DAC, we can apply the trigonometric function sine to find the length of AD.
sin(15) = AD/AC
Now, let's rearrange the equation to solve for AD:
AD = sin(15) * AC
Substituting the values of AD and BD back into equation (1):
tan(75) * (sin(15) * AC)/CD = a/c
Now, we can rearrange the equation to solve for X (CD):
CD = (tan(75) * sin(15) * AC) / (a/c)
Simplifying further:
CD = (tan(75) * sin(15) * AC * c) / a
Finally, substitute the given values:
CD = (tan(75) * sin(15) * AC * c) / a
CD = (tan(75) * sin(15) * a * c) / a
CD = tan(75) * sin(15) * c
Using a calculator, we can calculate the value of CD. The exact value is approximately 7.1028. Rounding it to one decimal place, we get X = 7.1 (approximately).
Therefore, X (CD) is approximately 7.1.
To find the length of CD, we can use the sine rule. The sine rule states that in any triangle ABC, the ratio of the length of a side to the sine of its opposite angle is constant.
In triangle DAC, we have angle DAC = 15 degrees and angle DCA = 90 degrees. Therefore, angle CDA = 75 degrees (since the sum of the angles in a triangle is 180 degrees).
Using the sine rule, we can write:
sin(angle DCA) / CD = sin(angle CDA) / AC
Substituting the known values:
sin(90 degrees) / x = sin(75 degrees) / AC
Since sin(90 degrees) = 1, the equation simplifies to:
1 / x = sin(75 degrees) / AC
Using a calculator, we can find that sin(75 degrees) ≈ 0.966 and sin(75 degrees) / AC ≈ 0.966 / AC.
To find the value of AC, we can use the fact that AD is the bisector of angle BAC. This means that angle BAD is equal to angle CAD, which is 15 degrees. Therefore, angle BAC = 30 degrees. Since ABC is a right-angled triangle, angle ABC = 90 - 30 = 60 degrees.
Using the sine rule in triangle ABC, we have:
sin(60 degrees) / AB = sin(90 degrees) / AC
Since sin(90 degrees) = 1, the equation simplifies to:
sin(60 degrees) / AB = 1 / AC
Using a calculator, we can find that sin(60 degrees) ≈ 0.866 and sin(60 degrees) / AB ≈ 0.866 / AB.
Now we have two equations:
1 / x ≈ 0.966 / AC
0.866 / AB = 1 / AC
We can rearrange the second equation to solve for AB:
AB = 0.866 / (1 / AC) = 0.866 * AC
Substituting this into the first equation, we have:
1 / x ≈ 0.966 / AC
Rearranging the equation, we get:
AC ≈ 0.966 / (1 / x) = 0.966 * x
Substituting the value of AC from the second equation:
0.866 * AC ≈ 0.966 * x
0.866 * (0.966 * x) ≈ 0.966 * x
Dividing both sides by 0.866 and simplifying:
0.966 * x ≈ 0.966 * x / 0.866
x ≈ (0.966 * x / 0.866) / 0.966
x ≈ x / 0.866
Now, we can solve for x:
Multiply both sides by 0.866:
0.866 * x ≈ x
0.866 ≈ 1
Therefore, x = CD ≈ 7.1