In△ABC, we knowt hatAB=5,BC=9, and CA=7. D is a point on BC such that AD bisects ∠BAC. What is DC?
To find the length of DC, we need to apply the angle bisector theorem.
According to the angle bisector theorem states that in a triangle, an angle bisector divides the opposite side into segments that are proportional to the lengths of the other two sides.
Here, AD is the angle bisector of ∠BAC. Let's denote the length of DC as x.
Using the angle bisector theorem, we know that:
AD/DC = AB/BC
Substituting the given values:
AD/x = 5/9
To find x, we can cross-multiply and solve for x:
AD * 9 = 5 * x
Now, we need to find the length of AD.
Since AD bisects ∠BAC, it divides ∠BAC into two equal angles, which means ∠DAB ≅ ∠DAC.
We can use the Law of Sines to find AD. The Law of Sines states that for any triangle △ABC with sides of lengths a, b, and c opposite to angles A, B, and C respectively:
a/sin(A) = b/sin(B) = c/sin(C)
Here, we have side AB = 5, side AC = 7, and opposite angles ∠ABC and ∠ACB.
Using the Law of Sines, we can find AD:
AD/sin(B) = AB/sin(∠DAB)
AD/sin(B) = 5/sin(∠DAB)
Similarly:
AD/sin(C) = AC/sin(∠DAC)
AD/sin(C) = 7/sin(∠DAC)
Since ∠DAB ≅ ∠DAC, we can write:
AD/sin(B) = AD/sin(C)
Cross-multiplying:
sin(B) = sin(C)
Since B and C are angles of a triangle, their sum is 180 degrees.
B + C = 180 degrees
So, we have:
sin(B) = sin(C) = sin(180 - B)
This means that angle B and angle 180 - B have the same sine value.
Since sine is positive for angles between 0 and 180 degrees, we can conclude that angle B = 180 - B.
Simplifying:
2B = 180
B = 90
So, angle B is 90 degrees.
Now, we can find AD using the Law of Sines:
AD/sin(90) = 5/sin(∠DAB)
AD/1 = 5/sin(∠DAB)
AD = 5/sin(∠DAB)
Since ∠DAB + ∠DAC = 180 degrees (opposite angles of a triangle), we have:
∠DAB = 180 - B - C
∠DAB = 180 - 90 - C
∠DAB = 90 - C
Substituting this into the equation for AD:
AD = 5/sin(90 - C)
Now, we can substitute AD back into the equation for the angle bisector to solve for x:
AD/x = 5/9
(5/sin(90 - C))/x = 5/9
Cross-multiplying:
9 * (5/sin(90 - C)) = 5 * x
45/sin(90 - C) = 5x
Since sin(90 - C) = cos(C), we can simplify:
45/cos(C) = 5x
Dividing both sides by 5:
9/cos(C) = x
So, the length of DC is 9/cos(C).