Question

Aavesh, a courier delivery agent, starts at point A and makes a delivery each at points B, C and D, in that order. He travels in a straight line between any two consecutive points. The following are known: (i) AB and CD intersect at a right angle at E, and; (ii) BC, CE and ED are respectively 1.3 km, 0.5 km and 2.5 km long. If AD is parallel to BC, then what is the total distance (in km) that Aavesh covers in travelling from A to D?

• A. 11.5
• B. 10.8
• C. 8.8
• D. 11.2
• E. 12.5

Hint:

In geometry problems with intersecting lines and parallels, carefully identify the correct similar triangles and use the given lengths to set up proportions.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Points A, B, C, D form a path. Segments AB and CD meet at E at $90^\circ$. $AD \parallel BC$.
Step 2: Key Formula or Approach:
1. Pythagoras theorem in $\triangle BEC$ to find $BE$.
2. Similar triangles $\triangle EBC \sim \triangle EAD$ (since $AD \parallel BC$ and vertically opposite angles are equal at E).
3. Total distance $= AB + BC + CD$.
Step 3: Detailed Explanation:
In $\triangle BEC$: $BE = \sqrt{BC^2 - CE^2} = \sqrt{1.3^2 - 0.5^2} = \sqrt{1.69 - 0.25} = 1.2$ km.
Since $\triangle EBC \sim \triangle EAD$, ratio of sides $= \frac{ED}{CE} = \frac{2.5}{0.5} = 5$.
So, $AE = 5 \times BE = 5 \times 1.2 = 6.0$ km.
$AB = AE + EB = 6.0 + 1.2 = 7.2$ km.
$CD = CE + ED = 0.5 + 2.5 = 3.0$ km.
Total Distance $= AB + BC + CD = 7.2 + 1.3 + 3.0 = 11.5$ km.
Step 4: Final Answer:
Total distance covered is 11.5 km.