Question

1. the two figures below are similar. solve for x and z.

Explanation:

Step1: Find the scale factor

The corresponding sides of similar figures are proportional. We can use the sides of length 54 mm and 40 mm? Wait, no, looking at the vertical sides? Wait, the first figure has a side of 54 mm and the second has 40 mm? Wait, maybe the sides with lengths 54 and 40? Wait, no, let's check the other sides. Wait, the first figure has a vertical side of 15 mm and the second has z? Wait, no, maybe the sides of length 33 (wait, the first figure has a segment of 33? Wait, the first figure: 54 mm, 15 mm, 33 mm? Wait, the second figure: 40 mm, z, and some other? Wait, maybe the ratio of corresponding sides. Let's assume that the sides of length 54 and 40 are corresponding? Wait, no, maybe 54 and 40? Wait, no, let's see. Wait, the first figure has a side of 54 mm, the second has 40 mm? Wait, no, maybe the vertical sides: 15 mm and z, and the slant sides: x and 40 mm, and the horizontal segment: 33 mm and... Wait, maybe the ratio is 54/40? Wait, no, let's do it properly.

Wait, similar figures have proportional sides. Let's identify corresponding sides. Let's say the first figure has a side of length 54 mm, and the second has 40 mm? Wait, no, maybe the side with length 33 mm (wait, the first figure has a horizontal segment of 33 mm? Wait, the first figure: 54 mm (vertical?), 15 mm (vertical?), 33 mm (horizontal?), and x (slant). The second figure: 40 mm (slant), z (vertical), and some horizontal? Wait, maybe the ratio of similarity is 54/40? Wait, no, let's check the vertical sides. Wait, the first figure has a vertical side of 15 mm, the second has z. The slant side of the first is x, the second is 40 mm. The horizontal segment of the first is 33 mm, the second is... Wait, maybe the ratio is 54/40? Wait, no, let's do it step by step.

Wait, maybe the sides of length 54 and 40 are corresponding? Wait, no, 54 and 40: 54/40 = 27/20. Wait, but the first figure has a vertical side of 15 mm, so z would be 15*(20/27)? No, that doesn't make sense. Wait, maybe I got the corresponding sides wrong. Wait, maybe the first figure's slant side is x, and the second's is 40 mm. The first figure's vertical side is 15 mm, the second's is z. The first figure's horizontal segment is 33 mm, the second's is... Wait, maybe the ratio is 54/40? Wait, no, let's look at the other sides. Wait, the first figure has a side of 54 mm, the second has 40 mm. So the scale factor from the first to the second is 40/54 = 20/27. Wait, no, scale factor is (length of second)/(length of first) if they are similar. Wait, similar figures: corresponding sides are in proportion. So if the first figure has a side of length a, the second has a side of length b, then a/b = k (scale factor from first to second) or b/a = k (from second to first).

Wait, let's assume that the side of length 54 mm in the first figure corresponds to the side of length 40 mm in the second figure? No, that would be a reduction, but maybe. Wait, no, maybe the vertical side of 15 mm in the first figure corresponds to z in the second, and the slant side x in the first corresponds to 40 mm in the second, and the horizontal segment 33 mm in the first corresponds to... Wait, maybe the ratio is 54/40 = 27/20. Wait, no, let's check the horizontal segment. Wait, the first figure has a horizontal segment of 33 mm? Wait, the first figure: 54 mm (let's say the left side), 15 mm (top vertical), 33 mm (top horizontal), and x (slant). The second figure: 40 mm (slant), z (top vertical), and some top horizontal. Wait, maybe the ratio is 54/40 = 27/20. So then, for the vertical side: 15/z = 27/20? No, that would be z = 15*(20/27) = 100/9 ≈ 11.11, which doesn't seem right. Wait, maybe I mixed up the correspondence.

Wait, maybe the first figure's slant side is x, and the second's is 40 mm, and the first figure's vertical side is 15 mm, the second's is z, and the first figure's horizontal segment is 33 mm, the second's is... Wait, maybe the ratio is 33/22? Wait, 33 and 22: 33/22 = 3/2. Ah, that makes sense! So the scale factor from the second figure to the first is 3/2, because 33 is 3/2 of 22 (22*(3/2)=33). So that's the ratio. So then, the vertical side of the first figure is 15 mm, which should be 3/2 of z (since first is larger than second, because 33>22). So 15 = (3/2)*z → z = 15*(2/3) = 10 mm. Wait, no, if the first figure is larger, then the scale factor from second to first is 3/2, so second's side * 3/2 = first's side. So for the vertical side: z*(3/2) = 15 → z = 15*(2/3) = 10 mm. For the slant side: 40*(3/2) = x → x = 60 mm? Wait, no, the first figure's slant side is x, the second's is 40 mm. So if scale factor is 3/2 (second to first), then x = 40*(3/2) = 60 mm. And the vertical side: z*(3/2) = 15 → z = 10 mm. Wait, but what about the other side: 54 mm in the first figure. Let's check: 40*(3/2)=60, but 54 is not 60. Wait, maybe I got the corresponding sides wrong. Wait, maybe the 54 mm side in the first figure corresponds to the 40 mm side in the second? No, 54 and 40: 54/40 = 27/20, which is not 3/2. Wait, but 33 and 22 is 3/2. So maybe the 54 mm side is not corresponding to 40 mm. Wait, maybe the 54 mm side is a different side. Wait, let's re-examine the figure.

Wait, the first figure: top horizontal is 33 mm, top vertical is 15 mm, right vertical is 54 mm? Wait, no, maybe the first figure has a right side of 54 mm, and the second figure has a right side of 40 mm? No, that doesn't match the 33/22 ratio. Wait, maybe the 33 mm and 22 mm are corresponding, so the scale factor is 33/22 = 3/2. So then, the vertical side of the first figure is 15 mm, which is 3/2 of z (second figure's vertical side), so z = 15*(2/3) = 10 mm. The slant side of the first figure is x, which is 3/2 of 40 mm (second figure's slant side), so x = 40*(3/2) = 60 mm. The right side of the first figure is 54 mm, which should be 3/2 of the second figure's right side. Let's check: 54*(2/3) = 36 mm. But the second figure's right side is 40 mm? No, that's a problem. Wait, maybe the 54 mm and 40 mm are not corresponding. Maybe the 54 mm is a different side. Wait, maybe the first figure's right side is 54 mm, and the second figure's right side is... Wait, maybe I made a mistake in identifying the corresponding sides.

Wait, let's start over. Similar figures: all corresponding linear dimensions are in proportion. So let's list the corresponding sides:

- First figure: horizontal segment = 33 mm, vertical segment = 15 mm, slant side = x, other vertical side = 54 mm.

- Second figure: horizontal segment = 22 mm, vertical segment = z, slant side = 40 mm, other vertical side =?

Wait, the horizontal segments: 33 and 22. So 33/22 = 3/2. So the scale factor (first to second) is 2/3, or second to first is 3/2.

So vertical segment: first is 15, second is z. So 15/z = 3/2 → z = 15*(2/3) = 10 mm.

Slant side: first is x, second is 40. So x/40 = 3/2 → x = 40*(3/2) = 60 mm.

Other vertical side: first is 54, second is y. So 54/y = 3/2 → y = 54*(2/3) = 36 mm. But the second figure's other vertical side is 40? Wait, no, maybe the 40 mm is the slant side, not the vertical. So that's correct. So x is 60 mm, z is 10 mm.

Step2: Verify the ratio

Check the ratios:

- Horizontal: 33/22 = 3/2

- Vertical: 15/10 = 3/2

- Slant: 60/40 = 3/2

- Other vertical: 54/36 = 3/2

All ratios are 3/2, so that's consistent.

Answer:

x = 60 mm, z = 10 mm