Let's analyze and solve the questions one by one based on the photo.

1. Calculate the length of PM

From the diagram and the given dimensions:

• $KL = 37 \, \text{cm}$
• $NO = 2 \, \text{cm}$
• $OP = 3 \, \text{cm}$
• $PK = 9 \, \text{cm}$

Using the trapezoidal properties or the logic provided in the problem, I will calculate the requested length of $PM$.

2. Find the length of BD

From the given:

• $AB = 24 \, \text{cm}$
• $BC = 10 \, \text{cm}$
• $CD = 8 \, \text{cm}$

I will use the triangle proportionality theorem or similarity rules to determine $BD$.

3. Width of the building in the photo

• Height of the building $= 8 \, \text{m}$
• In the photo, the height of the building shrinks to $6 \, \text{cm}$.
• Using a similar ratio, we can compute the width of the building in the photo.

Let me now calculate all of these values systematically. Calculations for the given questions

Question 1: PM

KL = 37 # cm NO = 2 # cm OP = 3 # cm PK = 9 # cm

PM is calculated as the remaining portion between KL and NO + OP + PK

PM = KL - (NO + OP + PK)

Question 2: BD

AB = 24 # cm BC = 10 # cm CD = 8 # cm

Use triangle similarity or the intersecting chord theorem: AB/BC = BD/CD

BD = (AB * CD) / BC

BD = (AB * CD) / BC

Question 3: Width of the building in the photo

actual_height = 8 # m photo_height = 6 # cm

Assume the actual width is unknown, let it be x

Using the proportionality: actual_height / photo_height = actual_width / photo_width

Let the photo_width be calculated (actual_width is 16 m)

actual_width = 16 # m photo_width = (photo_height / actual_height) * actual_width

PM, BD, photo_widthHere are the results of the calculations:

1. Length of $PM$: $23 \, \text{cm}$
2. Length of $BD$: $19.2 \, \text{cm}$
3. Width of the building in the photo: $12 \, \text{cm}$

Would you like a detailed explanation for any of these steps?

Further Questions:

1. How is the similarity theorem used to calculate lengths in intersecting lines?
2. Could trapezoidal properties be applied differently for solving $PM$?
3. What are the possible real-life applications of proportionality in scaling photos?
4. How can you verify the accuracy of the measurements given?
5. Are there alternate methods to solve question 2 (BD) using geometry?

Tip:

When solving proportional problems, always double-check the units to ensure consistency throughout your calculation!