Question

1. the figure shows three right triangles, each with its longest side on the same line.
2. complete the table.

triangle	vertical side length	horizontal side length
cde
fgh

what do you notice about the side lengths of your triangles?
compare this information with your partner. what do you notice?

Explanation:

Step1: Analyze Triangle ABC

From the graph, triangle ABC has vertical side length (BC) = 3 and horizontal side length (AB) = 4.

Step2: Analyze Triangle CDE

Triangle CDE: The vertical side (CD? Wait, no, DE is vertical. Wait, looking at the graph, DE is vertical with length 6, and CD is horizontal. Wait, let's re - check. For triangle CDE, the vertical side length (DE) is 6, and the horizontal side length (CD) can be found by the grid. From point C to D, the horizontal distance: since AB is 4, and from C to D, let's see the grid. Wait, actually, the ratio of vertical to horizontal should be consistent. Wait, ABC has vertical 3, horizontal 4. Then CDE: vertical side is DE = 6, horizontal side CD: let's see, from C to D, how many units? Since AB is 4, and the slope should be same. The slope of ABC is 3/4. So for CDE, vertical is 6, so horizontal should be 8? Wait, no, maybe I misread. Wait, the graph: point C to B is 3 (vertical), A to B is 4 (horizontal). Then point D is to the right of C, and E is above D. The vertical length from D to E is 6, so DE = 6. Then CD: let's count the grid. From C to D, the horizontal distance: if AB is 4 (from A to B), then from C to D, how many? Wait, maybe the horizontal side of CDE is 8? Wait, no, maybe I made a mistake. Wait, let's look at the table. Wait, the problem is to complete the table. Let's re - examine the graph:

Triangle ABC: vertical side (BC) = 3, horizontal side (AB) = 4.

Triangle CDE: vertical side (DE) = 6, horizontal side (CD): let's see, from C to D, the horizontal length. Since the line is the same, the ratio of vertical to horizontal should be 3/4. So if vertical is 6, horizontal is 8 (because 3/4 = 6/8).

Triangle FGH: vertical side (HG) = let's see, from G to H, the vertical length. The horizontal side (FG) = 2. Since the slope is 3/4, so vertical / 2 = 3/4 → vertical = 1.5? Wait, no, maybe the grid. Wait, looking at the graph, FG is 2 (horizontal), HG is vertical. Let's count the grid squares. If AB is 4 (4 grid units), BC is 3 (3 grid units). Then FG is 2 (2 grid units), so HG should be 1.5? No, maybe the vertical side of FGH is 1.5? Wait, no, maybe I misread. Wait, the graph: H is above G, and F is to the left of G. FG is 2 (horizontal), HG is vertical. Let's see the ratio. ABC: 3/4, FGH: vertical / 2 = 3/4 → vertical = 1.5. But maybe the problem is to get the lengths from the graph. Wait, maybe the vertical side of ABC is 3, horizontal 4. CDE: vertical 6, horizontal 8. FGH: vertical 1.5, horizontal 2? No, that can't be. Wait, maybe the vertical side of FGH is 1.5? No, maybe the graph has integer lengths. Wait, maybe I made a mistake. Wait, the problem says "three right triangles, each with its longest side on the same line". So they are similar triangles. So the ratios of corresponding sides are equal. So ABC: vertical = 3, horizontal = 4. CDE: vertical = 6 (since 3*2 = 6), so horizontal = 4*2 = 8. FGH: horizontal = 2 (since 4/2 = 2), so vertical = 3/2 = 1.5? No, that's not integer. Wait, maybe the vertical side of FGH is 1.5? But maybe the graph has different scaling. Wait, maybe the vertical side of FGH is 1.5 and horizontal 2, but that seems odd. Wait, let's check the graph again. The point F to G is 2 (horizontal), G to H is vertical. The line passes through F, H, E. So the slope is (H_y - F_y)/(G_x - F_x) = (H_y - F_y)/2. And the slope of ABC is (3 - 0)/(4 - 0)= 3/4. So (H_y - F_y)/2 = 3/4 → H_y - F_y = 1.5. So vertical side of FGH is 1.5, horizontal is 2.

But maybe the problem is to find the lengths as per the graph. Let's list:

Triangle ABC:
Vertical side length: 3
Horizontal side length: 4

Triangle CDE:
Vertical side length: 6 (since DE is 6, as per the graph, the vertical segment from D to E is 6)
Horizontal side length: 8 (since the ratio of vertical to horizontal is 3:4, 6/8 = 3/4)

Triangle FGH:
Vertical side length: 1.5 (since FG is 2, and 3/4 = vertical/2 → vertical = 1.5)
Horizontal side length: 2

Now, for the notice: The ratios of vertical side length to horizontal side length for each triangle are equal (3/4, 6/8 = 3/4, 1.5/2 = 3/4), so the triangles are similar (same slope, so similar right triangles).

Answer:

Triangle	Vertical side length	Horizontal side length
CDE	6	8
FGH	1.5	2

Notice: The ratio of vertical side length to horizontal side length is the same (3/4) for all three triangles, indicating they are similar right triangles (same slope, so corresponding angles are equal and sides are proportional).