Similar triangles with shadows (height–shadow proportion): A tree is 12 m tall and casts an 8 m long shadow at a given time. At the same time and under the same sun angle, a flagpole casts a 100 m long shadow. Using direct proportion of height to shadow length, what is the height of the flagpole?

Introduction / Context:
Objects standing vertically under the same sun angle form similar right triangles with the ground. Hence, height is directly proportional to shadow length. This is a straightforward unitary-method application grounded in geometric similarity.

Given Data / Assumptions:

• Tree height = 12 m
• Tree shadow = 8 m
• Flagpole shadow = 100 m
• Sun angle is the same for both, so triangles are similar and ratios hold.

Concept / Approach:
Height / shadow is constant across similar triangles. Compute the ratio from the tree and apply it to the flagpole’s shadow to find its height.

Step-by-Step Solution:
Height-to-shadow ratio = 12 / 8 = 1.5Flagpole height = 1.5 * 100 = 150 m

Verification / Alternative check:
Set up proportion: 12 : 8 = H : 100 → H = (12/8) * 100 = 150 m, which matches the ratio approach exactly.

Why Other Options Are Wrong:
200 m and 175 m assume a larger ratio; 125 m and 115 m assume smaller ratios, contradicting the given 12-to-8 relationship.

Common Pitfalls:
Using area or squaring ratios (not needed) instead of simple linear proportionality. Heights and shadows scale linearly, not quadratically, under similar-triangle conditions.

Final Answer:
150 m