In \( \triangle P Q R, P Q=4 \mathrm{~cm}, P R=8 \mathrm{~cm} \) and \( R T=6 \) cm . Find (i) the area of \( \triangle P Q R \) (ii) the length of QS.

To solve the problem, we need to extract the known conditions and then proceed step by step. ### Known Conditions: 1. In triangle \( PQR \): - \( PQ = 4 \, \text{cm} \) - \( PR = 8 \, \text{cm} \) - \( RT = 6 \, \text{cm} \) ### Step 1: Find the area of triangle \( PQR \) To find the area of triangle \( PQR \), we can use the formula for the area of a triangle when two sides and the included angle are known. However, we do not have the angle directly. Instead, we can use Heron's formula, which requires the semi-perimeter and the lengths of all three sides. We need to find the length of \( QR \) first. Assuming \( T \) is a point on \( QR \) such that \( RT = 6 \, \text{cm} \), we can denote \( QT = x \). Therefore, the length of \( QR \) can be expressed as: \[ QR = QT + RT = x + 6 \] Now, we can apply the triangle inequality to find the possible values for \( QR \): 1. \( PQ + QR > PR \) implies \( 4 + (x + 6) > 8 \) or \( x > -2 \) (always true since lengths are positive). 2. \( PQ + PR > QR \) implies \( 4 + 8 > (x + 6) \) or \( 12 > x + 6 \) or \( x < 6 \). 3. \( PR + QR > PQ \) implies \( 8 + (x + 6) > 4 \) or \( x > -10 \) (always true). Thus, \( 0 < x < 6 \). Now, we can calculate the area using Heron's formula. The semi-perimeter \( s \) is given by: \[ s = \frac{PQ + PR + QR}{2} = \frac{4 + 8 + (x + 6)}{2} = \frac{18 + x}{2} \] The area \( A \) can be calculated as: \[ A = \sqrt{s(s - PQ)(s - PR)(s - QR)} \] Substituting the values: \[ A = \sqrt{\frac{18 + x}{2} \left(\frac{18 + x}{2} - 4\right) \left(\frac{18 + x}{2} - 8\right) \left(\frac{18 + x}{2} - (x + 6)\right)} \] ### Step 2: Find the length of \( QS \) To find \( QS \), we need more information about point \( S \). If \( S \) is the foot of the altitude from \( P \) to \( QR \), we can use the area calculated above to find \( QS \). The area can also be expressed as: \[ A = \frac{1}{2} \times base \times height \] where the base is \( QR \) and the height is the length from \( P \) to line \( QR \). However, without additional information about the position of point \( S \) or the angle measures, we cannot determine \( QS \) directly. ### Conclusion To summarize: 1. We need to calculate the area of triangle \( PQR \) using Heron's formula, but we need the length of \( QR \) which depends on \( x \). 2. The length of \( QS \) cannot be determined without additional information about point \( S \). If you have any additional information or constraints regarding point \( S \) or the angle measures, please provide that for a more precise solution.