Fractals and Visualising Solids | FIO

Question 10

Observe the front view, top view and side view of the different lines in Fig. 4.6. Is there any relation between their lengths?

Question diagram 1
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Solution
Understand the Question
  • Orthographic views (front, top, side) show an object's projection onto perpendicular planes:
    • Front View: Shows horizontal width (left-to-right) and vertical height (up-to-down).
    • Top View: Shows horizontal width (left-to-right) and depth (front-to-back).
    • Side View: Shows vertical height (up-to-down) and depth (front-to-back).
  • For lines lying at a constant height (horizontal in front view), the actual 3D length is determined by combining the left-to-right component and the depth component: Actual Length=(Lleft-right)2+(Ldepth)2\text{Actual Length} = \sqrt{(L_{\text{left-right}})^2 + (L_{\text{depth}})^2}
  • Since all three lines have identical front-view lengths, comparing their depths directly determines the relation between their actual lengths.

Observe the front view, top view and side view of the different lines in Fig. 4.6. Is there any relation between their lengths?

Step 1 · Analyze the Orthographic Views of Each Line

Question diagram

  • Line (a):

    • Front View: Horizontal line   ⟹  \implies no change in height.
    • Top View: Horizontal line   ⟹  \implies no depth component.
    • Side View: A single dot   ⟹  \implies depth is 00.

  • Line (b):

    • Front View: Horizontal line of the same length as (a)   ⟹  \implies same width, no change in height.
    • Top View: Diagonal line   ⟹  \implies extends in both width and depth.
    • Side View: Short horizontal line   ⟹  \implies has depth LdepthbL_{\text{depth}_b}.

  • Line (c):

    • Front View: Horizontal line of the same length as (a) and (b)   ⟹  \implies same width, no change in height.
    • Top View: Diagonal line longer than (b)   ⟹  \implies extends in both width and depth.
    • Side View: Longer horizontal line   ⟹  \implies has depth LdepthcL_{\text{depth}_c}.

Step 2 · Compare Length Components

All three lines have the same left-to-right component in the front view: Lleft-right=constantL_{\text{left-right}} = \text{constant}

Comparing the front-to-back (depth) components from the side views:

  • Line (a)(a): Ldeptha=0L_{\text{depth}_a} = 0
  • Line (b)(b): Ldepthb>0L_{\text{depth}_b} > 0
  • Line (c)(c): Ldepthc>LdepthbL_{\text{depth}_c} > L_{\text{depth}_b}

Therefore: Ldepthc>Ldepthb>0L_{\text{depth}_c} > L_{\text{depth}_b} > 0

Step 3 · Determine the Relation Between Actual Lengths

The actual length of each line is calculated using the Pythagorean relation: Actual Length=(Lleft-right)2+(Ldepth)2\text{Actual Length} = \sqrt{(L_{\text{left-right}})^2 + (L_{\text{depth}})^2}

For each line:

Actual Length (a)=(Lleft-right)2+02=Lleft-rightActual Length (b)=(Lleft-right)2+(Ldepthb)2Actual Length (c)=(Lleft-right)2+(Ldepthc)2\begin{aligned} \text{Actual Length (a)} &= \sqrt{(L_{\text{left-right}})^2 + 0^2} = L_{\text{left-right}} \\[0.6em] \text{Actual Length (b)} &= \sqrt{(L_{\text{left-right}})^2 + (L_{\text{depth}_b})^2} \\[0.6em] \text{Actual Length (c)} &= \sqrt{(L_{\text{left-right}})^2 + (L_{\text{depth}_c})^2} \end{aligned}

Since Ldepthc>Ldepthb>0L_{\text{depth}_c} > L_{\text{depth}_b} > 0 and Lleft-rightL_{\text{left-right}} is equal for all three: Actual Length (a)<Actual Length (b)<Actual Length (c)\text{Actual Length (a)} < \text{Actual Length (b)} < \text{Actual Length (c)}

Diagram 1

Answer

Yes, there is a relation between their lengths:

Length(a)<Length(b)<Length(c)\text{Length}(a) < \text{Length}(b) < \text{Length}(c)

Line (a) is the shortest and line (c) is the longest.

Common Mistakes
  • Judging Length by Front View Alone: Assuming all three lines are equal in length because their front views are identical. A 2D projection does not capture extension along the depth axis.
  • Misinterpreting Side View Dot: Overlooking that a dot in the side view indicates zero depth, which means the line is oriented purely parallel to the front plane.

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Q10

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Solids to choose from:

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