in the triangle ABC,AB=3 BC =4 CA=2.D is the point on CA such that AD bisects angle A.M is the mid point of CA.F is a point on AB such that angle AFD=90. DF and am meet at the point E.Then AF:FB=?,DE:EF=?
To find the ratios AF:FB and DE:EF in the given triangle ABC, we need to use some properties of angles and triangle similarity.
1. Let's start by finding the length of AD and DC. Since AD bisects angle A, we can determine AD using the Angle Bisector Theorem. According to this theorem, AD/DC = AB/BC. Substituting the given values, we get AD/DC = 3/4.
2. Using the lengths of AB and BC, we can determine the length of AC using the Triangle Inequality Theorem. AC < AB + BC. Substituting the given values, we have AC < 3 + 4, which gives AC < 7.
3. Since M is the midpoint of AC, we can determine the length of AM by dividing AC in half. AM = AC/2 = 2/2 = 1.
4. Since AFD is a right angle, according to the Pythagorean Theorem, we have AF^2 + FD^2 = AD^2. Since we know AD/DC = 3/4, we can write AD = (3/7) * AC and DC = (4/7) * AC. Substituting these values, we have AF^2 + FD^2 = (3/7)^2 * AC^2.
5. Since F is a point on AB and angle AFD = 90, we can infer that angle AFB is also 90 degrees. Therefore, triangle AFB is a right-angled triangle.
6. Using the Pythagorean Theorem in triangle AFB, we have AF^2 + FB^2 = AB^2. Substituting the given values, we have AF^2 + FB^2 = 3^2.
7. Solving equations (4) and (6) together, we have (3/7)^2 * AC^2 + FB^2 = 3^2.
8. Since AD bisects angle A, we know that DE is parallel to BF (using alternate interior angles formed by transversal AM). Hence, triangles DEF and FBA are similar.
9. Using triangle similarity, we can write DE/EF = AD/FB. Substituting the values we know, we have DE/EF = (3/7) * AC / FB.
10. Since M is the midpoint of AC, we know that CM = AM = 1. Therefore, CD = AC - AD - AM = 7 - 3 - 1 = 3.
11. With the values we know, we can use the Triangle Similarity Theorem to write FB/AB = CD/BC. Substituting the given values, we have FB/3 = 3/4.
12. Combining equations (7) and (11), we can find AF:FB and DE:EF:
AF^2 / (3/7)^2 * AC^2 = FB^2 / (3/4)^2, and
DE/EF = (3/7) * AC / FB.
Using these equations, we can solve for the ratios AF:FB and DE:EF.