In `Delta P QR. PS bot QR. PS = 11 cm and ` area of `Delta PQR = 77 sq. cmIf area of `Delta PSQ: ` area of `Delta PSR = 3:4. ` find `QS and SR.`

Answer

To solve the problem step by step, we will follow the information given and apply the concepts of area in triangles.

### Step 1: Understand the given information
We have triangle PQR with PS perpendicular to QR. We know:
- PS = 11 cm (height)
- Area of triangle PQR = 77 sq. cm
- The ratio of the areas of triangles PSQ and PSR = 3:4

### Step 2: Calculate the total ratio
The total ratio of the areas of triangles PSQ and PSR is:
- 3 (for PSQ) + 4 (for PSR) = 7

### Step 3: Find the value of one part of the ratio
Since the total area of triangle PQR is 77 sq. cm, we can find the value of one part of the ratio:
- One part = Total Area / Total Ratio = 77 / 7 = 11 sq. cm

### Step 4: Calculate the areas of triangles PSQ and PSR
Now, we can calculate the areas of triangles PSQ and PSR:
- Area of triangle PSQ = 3 * 11 = 33 sq. cm
- Area of triangle PSR = 4 * 11 = 44 sq. cm

### Step 5: Use the area formula for triangle PSQ
The area of triangle PSQ can be expressed using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the known values:
\[ 33 = \frac{1}{2} \times QS \times 11 \]

### Step 6: Solve for QS
Rearranging the equation:
\[ 33 = \frac{11}{2} \times QS \]
Multiply both sides by 2:
\[ 66 = 11 \times QS \]
Now, divide by 11:
\[ QS = \frac{66}{11} = 6 \text{ cm} \]

### Step 7: Use the area formula for triangle PSR
Similarly, for triangle PSR:
\[ 44 = \frac{1}{2} \times SR \times 11 \]

### Step 8: Solve for SR
Rearranging the equation:
\[ 44 = \frac{11}{2} \times SR \]
Multiply both sides by 2:
\[ 88 = 11 \times SR \]
Now, divide by 11:
\[ SR = \frac{88}{11} = 8 \text{ cm} \]

### Final Answer
- QS = 6 cm
- SR = 8 cm

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