In \(\triangle PQR\), \(|PO| = 15\) cm, \(|OR| = 16\) cm, \(|RP| = 9\) cm. The bisector of \(\angle P\) meets \(OR\) at \(A\) and \(M\) is the midpoint of \(OR\). Calculate \(|AM|\).

In △ PQR , we are given ∣ PO ∣ = 15 cm, ∣ OR ∣ = 16 cm, and ∣ RP ∣ = 9 cm. We are tasked with finding ∣ A M ∣ , where A is the point where the angle bisector of ∠ P meets OR , and M is the midpoint of ∣ OR ∣ .
Steps:

Find the length of ∣ OM ∣ : Since M is the midpoint of ∣ OR ∣ , the length ∣ OM ∣ is half the length of ∣ OR ∣ . Therefore, ∣ OM ∣ = 2 ∣ OR ∣ ​ = 2 16 ​ = 8 cm

Apply the Angle Bisector Theorem: The Angle Bisector Theorem states that if a point A lies on the angle bisector and divides the opposite side ∣ OR ∣ , then the ratio of the segments is equal to the ratio of the adjacent sides. Hence, ∣ A R ∣ ∣ O A ∣ ​ = ∣ PR ∣ ∣ PO ∣ ​ = 9 15 ​ = 3 5 ​

Express ∣ O A ∣ and ∣ A R ∣ in terms of x : Let ∣ O A ∣ = 5 k and ∣ A R ∣ = 3 k , where k is a constant. Since ∣ OR ∣ = ∣ O A ∣ + ∣ A R ∣ , we have 5 k + 3 k = 16 8 k = 16 k = 2

Calculate ∣ O A ∣ : Substituting k = 2 , we find ∣ O A ∣ : ∣ O A ∣ = 5 × 2 = 10 cm

Find ∣ A M ∣ : Finally, ∣ A M ∣ = ∣ OM ∣ − ∣ O A ∣ . Thus, ∣ A M ∣ = 8 − 10 = − 2 cm However, since distance cannot be negative, ∣ A M ∣ is the absolute difference, i.e., ∣ A M ∣ = ∣ − 2∣ = 2 cm


Therefore, the length of ∣ A M ∣ is 2 cm.

Answered by DanielJosephParker | 2025-07-06