ABC is 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 do not know how to do the actual sum. Can you please help.
I do not know how too
if only angle given
There is something missing here.
1. where is the right-angle? (I assumed it was B)
2. some information about the length of sides has to be given.
(I assumed your answer of 7.1 for DC was correct and worked backwards. BC would have to be 13.25 )
Sure! To find the value of CD (denoted as X), we can use the angle bisector theorem in the triangle ABC.
According to the angle bisector theorem, the ratio of the lengths of the two sides that the angle bisector divides is equal to the ratio of the lengths of the other two sides. In this case, we have:
AC/AD = BC/BD
Substituting the given values, we have:
AC/AD = BC/BD
AC/X = BC/BD
Now, let's focus on the side lengths that we know:
AC = AD + CD
BC = BD + CD
Since AD is the angle bisector of angle BAC, we can use the fact that the angle bisector divides the opposite side in the same ratio as the two adjacent sides. This means that:
BD/DC = BA/AC
Substituting the known values:
BD/X = BA/(AD + CD)
We can simplify further:
BD/X = BA/(AD + X)
Now, let's consider the given information: Angle DAC = 15 degrees. Since AD is the angle bisector, we know that angle BAD is also 15 degrees (as angle BAD and angle DAC are opposite angles). Therefore, we can replace BA in the equation above with BC:
BD/X = BC/(AD + X)
Now, we have an equation with two unknowns: X and BD. To solve for X, we need to find the value of BD.
In a right-angled triangle, the angle bisector divides the side opposite the right angle into two segments that are proportional to the adjacent sides. Applying this property, we have:
BD/CD = AB/BC
Since it is a right-angled triangle, we know that AB = BC. Substituting this in the equation above:
BD/CD = AB/AB
BD/CD = 1
Rearranging the equation, we get:
BD = CD
Now, we can substitute BD = CD in the previous equation:
BD/X = BC/(AD + X)
CD/X = BC/(AD + X)
X/X = BC/(AD + X)
1 = BC/(AD + X)
Cross-multiplying:
AD + X = BC
Since AD = CD + X, we can substitute:
CD + X + X = BC
CD + 2X = BC
Now, substitute X = CD into the equation:
CD + 2CD = BC
3CD = BC
Now, we need the value of BC to find X. To find BC, we will use the sine rule in triangle ABC. The sine rule states:
BC/sin(B) = AC/sin(A)
Substituting the known values:
BC/sin(90) = AC/sin(15)
Since sin(90) = 1:
BC = AC/sin(15)
Substituting AC = AD + CD:
BC = (AD + CD)/sin(15)
Now we have BC in terms of AD and CD. Substituting this back into the equation for CD + 2X = BC:
CD + 2CD = (AD + CD)/sin(15)
3CD = (AD + CD)/sin(15)
Multiplying both sides by sin(15):
3CD * sin(15) = AD + CD
Now, we can substitute the given value for angle DAC = 15 degrees:
3CD * sin(15) = AD + CD
3CD * 0.2588 = AD + CD
0.7764CD = AD + CD
0.7764CD - CD = AD
0.7764CD - 1CD = AD
0.7764CD = AD
0.7764CD = CD + X
Substituting the given answer, X = 7.1:
0.7764CD = CD + 7.1
Now, we can solve this equation to find the value of CD:
0.7764CD - CD = 7.1
-0.2236CD = 7.1
Dividing both sides by -0.2236:
CD = 7.1 / -0.2236
CD ≈ -31.74
Since lengths cannot be negative, we discard the negative value.
Therefore, the value of CD (X) is not 7.1, but approximately 31.74.