Question 10

Orthographic views (front, top, side) help us understand the true shape and size of objects in three dimensions.

Step 1 — Analyze the views of each line

Let us consider the three lines, (a), (b), and (c), from top to bottom in the diagram. The Front View shows how much a line extends from left to right, and how much it extends up and down. The Top View shows how much a line extends from left to right, and how much it extends from front to back (its depth). The Side View shows how much a line extends up and down, and how much it extends from front to back.

Let's look at line (a): Its Front View is a horizontal line. This means it has no change in its up-and-down position. Its Top View is also a horizontal line. This means it has no change in its front-to-back depth. Its Side View is a single dot. This confirms it has no change in its up-and-down position and no change in its front-to-back depth. So, line (a) is a straight line that only extends from left to right. Its actual length is just its left-to-right component.

Let's look at line (b): Its Front View is a horizontal line, just like line (a). This means it also has no change in its up-and-down position. Its Top View is a diagonal line. This means it extends both from left to right and from front to back. Its Side View is a short horizontal line. Since we know it has no change in its up-and-down position (from the front view), this short horizontal line must represent how much it extends from front to back. So, line (b) extends from left to right and from front to back, but stays at the same height.

Let's look at line (c): Its Front View is a horizontal line, just like (a) and (b). This means it also has no change in its up-and-down position. Its Top View is a diagonal line, similar to (b) but appears longer. This means it extends both from left to right and from front to back. Its Side View is a longer horizontal line than for (b). Since it has no change in its up-and-down position, this longer horizontal line must represent how much it extends from front to back. So, line (c) also extends from left to right and from front to back, but stays at the same height.

Step 2 — Compare the components of length

From the Front Views, we observe that all three lines (a), (b), and (c) have the same length when viewed from the front. This means their "left-to-right" component is the same for all. Let us call this common length $L_{\text{left-right}}$.

Now, let's compare their "front-to-back" components using the Side Views: For line (a), the Side View is a dot. This means its "front-to-back" component is zero. For line (b), the Side View is a short horizontal line. Let its "front-to-back" component be L_{\text{depth_b}}. For line (c), the Side View is a longer horizontal line. Let its "front-to-back" component be L_{\text{depth_c}}.

By looking at the diagram, we can clearly see that the side view of (c) is longer than the side view of (b). So, we have the relation: L_{\text{depth_c}} > L_{\text{depth_b}} And since line (a) has no depth component: L_{\text{depth_c}} > L_{\text{depth_b}} > 0

Step 3 — Determine the relation between their actual lengths

Since none of the lines change their up-and-down position (their front views are horizontal), their actual length is found by combining their "left-to-right" component and their "front-to-back" component. This is like finding the hypotenuse of a right-angled triangle. The actual length of a line is given by $\sqrt{(L_{\text{left-right}})^2 + (L_{\text{depth}})^2}$.

Let's find the actual length for each line: For line (a): $\text{Actual Length (a)} = \sqrt{(L_{\text{left-right}})^2 + (0)^2}$ $= L_{\text{left-right}}$ For line (b): \text{Actual Length (b)} = \sqrt{(L_{\text{left-right}})^2 + (L_{\text{depth_b}})^2} For line (c): \text{Actual Length (c)} = \sqrt{(L_{\text{left-right}})^2 + (L_{\text{depth_c}})^2}

Since $L_{\text{left-right}}$ is the same for all three lines, and we know L_{\text{depth_c}} > L_{\text{depth_b}} > 0: The actual length of line (a) is the smallest. The actual length of line (b) is greater than line (a). The actual length of line (c) is the greatest. So, line (a) is the shortest, and line (c) is the longest.

Now, let's look at the Top Views in the diagram: The Top View of line (a) is a horizontal line of length $L_{\text{left-right}}$. The Top View of line (b) is a diagonal line of length \sqrt{(L_{\text{left-right}})^2 + (L_{\text{depth_b}})^2}. The Top View of line (c) is a diagonal line of length \sqrt{(L_{\text{left-right}})^2 + (L_{\text{depth_c}})^2}.

Notice that the lengths shown in the Top Views are exactly the actual lengths of the lines. By visually comparing the lengths of the lines in the Top View column of the diagram: The Top View of (a) is the shortest. The Top View of (b) is longer than (a). The Top View of (c) is longer than (b). This confirms our finding that line (a) is the shortest and line (c) is the longest.

Answer

Yes, there is a relation between their lengths. Top views show that (a) is the shortest, and (c) is the longest.