In a triangle PQR, QR = 12, RP = 16 and PQ = 20. M is a point on PQ such that MP:MQ = λ : 1 ( λ > 1 ) and the line RM divides angle PRQ in the ratio 2 : 1
then find the value of λ

To solve the problem involving triangle PQR with the given side lengths and the point M on PQ, we can use the concept of angle bisectors and the properties of triangles. The key here is to apply the angle bisector theorem, which states that the ratio of the lengths of the two segments created by the bisector is equal to the ratio of the other two sides of the triangle.

Understanding the Triangle and Given Information

We have triangle PQR with the following sides:

• QR = 12
• RP = 16
• PQ = 20

Point M lies on side PQ such that the ratio MP:MQ = λ:1, where λ > 1. The line RM divides angle PRQ in the ratio 2:1.

Applying the Angle Bisector Theorem

According to the angle bisector theorem, if a point R divides the angle PRQ into two segments, the lengths of the segments on the opposite side (PQ) will be proportional to the lengths of the other two sides of the triangle. In our case, we can express this as:

Let MP = λx and MQ = x, where MP + MQ = PQ = 20. Thus, we have:

λx + x = 20

(λ + 1)x = 20

From this, we can express x as:

x = 20 / (λ + 1)

Finding the Lengths of MP and MQ

Now substituting x back, we find:

• MP = λ(20 / (λ + 1)) = 20λ / (λ + 1)
• MQ = 20 / (λ + 1)

Setting Up the Ratio from the Angle Bisector

Since RM divides angle PRQ in the ratio 2:1, we can set up the following relationship using the sides opposite the angles:

According to the angle bisector theorem:

MP / MQ = RP / QR

Substituting the values we have:

(20λ / (λ + 1)) / (20 / (λ + 1)) = 16 / 12

This simplifies to:

λ = (16 / 12) = (4 / 3)

Cross-Multiplying to Solve for λ

Now, we can cross-multiply to eliminate the fractions:

20λ = (20 * 4) / 3

20λ = 80 / 3

λ = (80 / 3) / 20 = 4 / 3

Final Value of λ

Thus, the value of λ is:

λ = 4/3

This means that the point M divides the segment PQ in the ratio of 4:1, which is consistent with the condition that λ is greater than 1. This solution effectively uses the properties of triangles and the angle bisector theorem to arrive at the answer.