In a `DeltaPQR`, points M and N are on the sides PQ and PR respectively such that PM 0.6. PQ and `NR= 0.4PR`. What percentage of the area of the triangle PQR does that of triangle PMN form?

Options

A. A)`60%`
B. B)`50%`
C. C)`36%`
D. D)`55%`

Answer

To solve the problem, we need to find the percentage of the area of triangle PMN in relation to the area of triangle PQR. Let's break it down step by step.

### Step 1: Understand the dimensions of the triangles
Given:
- PM = 0.6 * PQ
- NR = 0.4 * PR

Assuming PQ = PR = 10 cm (as per the example in the video).

Calculating the lengths:
- PM = 0.6 * 10 cm = 6 cm
- MQ = PQ - PM = 10 cm - 6 cm = 4 cm
- NR = 0.4 * 10 cm = 4 cm
- NR = PR - NR = 10 cm - 4 cm = 6 cm

### Step 2: Calculate the area of triangle PQR
The area of triangle PQR can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

For triangle PQR:
- Base = PQ = 10 cm
- Height = PR = 10 cm (assuming the height is the same as the length of PR for simplicity)

Thus, the area of triangle PQR is:
\[ \text{Area}_{PQR} = \frac{1}{2} \times 10 \times 10 = 50 \, \text{cm}^2 \]

### Step 3: Calculate the area of triangle PMN
For triangle PMN:
- Base = PM = 6 cm
- Height = PN (which is also 6 cm, as NR = 4 cm and the total height remains the same)

Thus, the area of triangle PMN is:
\[ \text{Area}_{PMN} = \frac{1}{2} \times 6 \times 6 = 18 \, \text{cm}^2 \]

### Step 4: Calculate the percentage of the area of triangle PMN to triangle PQR
To find the percentage:
\[ \text{Percentage} = \left( \frac{\text{Area}_{PMN}}{\text{Area}_{PQR}} \right) \times 100 \]

Substituting the areas we calculated:
\[ \text{Percentage} = \left( \frac{18}{50} \right) \times 100 = 36\% \]

### Conclusion
The area of triangle PMN forms 36% of the area of triangle PQR.

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