Let's work through the question step-by-step to find the true length of the line.
### Question 11:
Given:
- The length of the top view of the line (parallel to the V.P. and inclined at 45° to the H.P.) is 50 mm.
- One end of the line is 12 mm above the H.P.
- The same end of the line is 25 mm in front of the V.P.
We need to determine the true length of the line.
#### Steps:
1. Understand the given dimensions and position:
- The top view (horizontal projection) of the line is given as 50 mm. This means when viewing the line from above, its length appears to be 50 mm.
- The inclination angle of the line with the horizontal plane (H.P.) is 45°.
2. Use of trigonometric relationships:
- To find the true length of the line, we need to consider the actual orientation of the line in 3D space.
- Since the line is inclined at 45° to the H.P., we can use trigonometry to adjust the projected length (top view length) to find the true length.
3. Application of the cosine function:
- The formula to find the true length ([tex]\( L_{true} \)[/tex]) when the inclined length [tex]\( L_{top view} \)[/tex] and angle ([tex]\( \theta \)[/tex]) are known is:
[tex]\[
L_{true} = \frac{L_{top view}}{\cos{\theta}}
\][/tex]
- Here, [tex]\( \theta = 45° \)[/tex] and [tex]\( L_{top view} = 50 \)[/tex] mm.
4. Substitute the values:
- Compute the cosine of 45°:
[tex]\[
\cos{45°} = \frac{\sqrt{2}}{2} \approx 0.707
\][/tex]
- Substitute the values into the formula and solve:
[tex]\[
L_{true} = \frac{50 \text{ mm}}{0.707}
\][/tex]
[tex]\[
L_{true} \approx 70.7107 \text{ mm}
\][/tex]
Thus, the true length of the line is approximately 70.71 mm.
This is the detailed step-by-step solution to determine the true length of the line when its top view length is 50 mm and it is inclined at 45° to the H.P with one end 12 mm above the H.P and 25 mm in front of the V.P.
For question 12, please provide the complete problem statement, and we will solve it accordingly.
