If ABC is a triangle with AB=20,BC=22 and CA=24. Let D lie on BC such that AD is the angle bisector of ∠BAC. What is AD^2?
This is a brilliant qn so i will only give you a hint: Use the law of sines.
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http://www.jiskha.com/display.cgi?id=1371989906
To find AD^2, we can use the Angle Bisector Theorem and the Law of Cosines.
1. Angle Bisector Theorem:
According to the Angle Bisector Theorem, the ratio of the lengths of the sides of a triangle to the lengths of the segments it creates on the opposite side is equal to the ratio of the lengths of the other two sides.
In this case, we have AB = 20, BC = 22, and CA = 24. Let D be the point on BC that divides it into BD and DC. Applying the Angle Bisector Theorem, we have:
AB/BD = AC/CD
2. Law of Cosines:
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles and the length of the opposite side. In our case, we can use the Law of Cosines to express the cosine of angle BAC in terms of the lengths of the sides of the triangle.
Applying the Law of Cosines to triangle ABC and using the fact that AD is the angle bisector of angle BAC, we have:
cos(BAC) = (BC^2 + AC^2 - AB^2) / (2 * BC * AC)
Now, let's solve the problem:
1. Calculate BD and CD using the Angle Bisector Theorem:
Since AB/BD = AC/CD, we can set up the following equation:
20/BD = 24/CD
Cross-multiplying gives us:
20 * CD = 24 * BD
2. Calculate the cosine of angle BAC using the Law of Cosines:
cos(BAC) = (BC^2 + AC^2 - AB^2) / (2 * BC * AC)
cos(BAC) = (22^2 + 24^2 - 20^2) / (2 * 22 * 24)
Simplifying:
cos(BAC) = (484 + 576 - 400) / (2 * 22 * 24)
cos(BAC) = 660 / 1056
cos(BAC) = 5/8
3. Use the cosine of angle BAC to find AD^2:
Using the Angle Bisector Theorem, we know that BD/CD = AB/AC.
Let's substitute the values we already know:
BD/CD = AB/AC
BD/CD = 20/24
BD/CD = 5/6
From step 1, we found that 20 * CD = 24 * BD. Using this equation and the ratio BD/CD = 5/6, we can solve for BD and CD.
24 * BD = 20 * CD
24 * BD = 20 * (6/5) * BD
24 * BD = 24 * BD
Since both sides are equal, we can say that BD = CD = x.
Now, we can use the Law of Cosines again to find AD^2.
cos(BAC) = (BC^2 + AC^2 - AB^2) / (2 * BC * AC)
5/8 = (22^2 + 24^2 - x^2) / (2 * 22 * 24)
Simplifying and solving for x^2:
5/8 = (484 + 576 - x^2) / 1056
5/8 = (1060 - x^2) / 1056
Cross-multiplying:
5 * 1056 = 8 * (1060 - x^2)
5280 = 8480 - 8x^2
8x^2 = 3200
x^2 = 400
Therefore, AD^2 = x^2 = 400.