Question

1. for each scaled copy, write the scale factor that takes triangle t to that triangle. leave blank if it is not a scaled copy.

triangle\tscale factor
a\t
b\t
c\t
d\t
e\t
f\t

triangle diagrams: a (5, 6, 4); b (4, 5, 3); c (5, 6.4, 4); d (6, 7.5, 4.5); e (6, 10, 8); f (6, 8, 7)

Answer:

To solve for the scale factor of each triangle (assuming Triangle \( T \) has sides, let's first identify the sides of Triangle \( T \). Wait, maybe Triangle \( T \) is one of them? Wait, looking at the triangles:

Let's assume Triangle \( T \) is, say, Triangle \( A \) with sides 5, 6, 4? Wait, no, maybe we need to find which triangles are scaled copies. Wait, maybe Triangle \( T \) is the one with sides 5, 6, 4? Wait, let's check each triangle:

First, let's list the sides of each triangle:

• Triangle \( A \): 5, 6, 4 (let's check the order, maybe 4, 5, 6? Wait, the triangle has sides 4, 5, 6? Wait, the first triangle \( A \) has sides 5, 6, 4 (base 4, other sides 5 and 6? Wait, maybe the sides are 4, 5, 6. Let's confirm:

• Triangle \( A \): sides 4, 5, 6? Wait, the image shows Triangle \( A \) with sides 5, 6, 4 (maybe 4, 5, 6 as a right triangle? No, 4-5-6 is not a right triangle. Wait, maybe Triangle \( E \) is 6, 8, 10 (which is a right triangle, 6-8-10, scale factor 2 from 3-4-5). Wait, Triangle \( B \) is 3, 4, 5 (right triangle). Ah! So maybe Triangle \( T \) is Triangle \( B \) with sides 3, 4, 5. Then we can find the scale factor for each triangle relative to \( B \) (3,4,5).

Let's check each triangle:

1. Triangle \( A \): sides 4, 5, 6? Wait, no, the first triangle \( A \) (top left) has sides 5, 6, 4? Wait, the image: Triangle \( A \) has sides 5, 6, 4 (maybe 4, 5, 6). Wait, no, let's look at the numbers:

• Triangle \( B \): 3, 4, 5 (sides 3, 4, 5)
• Triangle \( A \): 4, 5, 6? Wait, no, the first triangle \( A \) (top left) has 5, 6, 4 (maybe 4, 5, 6). Wait, no, let's check the other triangles:

• Triangle \( C \): 4, 5, 6.4? Wait, no, Triangle \( C \) has sides 4, 5, 6.4? Wait, no, the numbers: Triangle \( C \) has 4, 5, 6.4? Wait, maybe I misread. Let's list all triangles with their sides:

• Triangle \( B \): 3, 4, 5 (sides 3, 4, 5)
• Triangle \( A \): 4, 5, 6 (sides 4, 5, 6)
• Triangle \( C \): 4, 5, 6.4 (sides 4, 5, 6.4)
• Triangle \( D \): 4.5, 6, 7.5 (sides 4.5, 6, 7.5)
• Triangle \( E \): 6, 8, 10 (sides 6, 8, 10)
• Triangle \( F \): 6, 7, 8 (sides 6, 7, 8)

Now, let's assume Triangle \( T \) is Triangle \( B \) (3,4,5). Then the scale factor \( k \) for a scaled copy is \( \frac{\text{side of copy}}{\text{side of } T} \). Let's check each:

• Triangle \( A \): sides 4, 5, 6. Let's check with \( B \) (3,4,5). \( \frac{4}{3} \approx 1.333 \), \( \frac{5}{4} = 1.25 \), \( \frac{6}{5} = 1.2 \). Not consistent. So maybe \( T \) is another triangle.

Wait, Triangle \( E \) is 6,8,10. Let's see: 6-8-10 is 2*(3-4-5). So 3-4-5 scaled by 2 is 6-8-10. So maybe \( T \) is 3-4-5 (Triangle \( B \)), then \( E \) is scale factor 2.

Triangle \( D \): 4.5, 6, 7.5. Let's check: 4.5/3 = 1.5, 6/4 = 1.5, 7.5/5 = 1.5. So scale factor 1.5.

Triangle \( A \): 4,5,6. Let's check with 3-4-5: 4/3 ≈1.333, 5/4=1.25, 6/5=1.2. Not consistent. Wait, maybe \( T \) is 4-5-6? No, 4-5-6 is not a right triangle. Wait, Triangle \( A \) has sides 5,6,4 (maybe 4,5,6). Let's check Triangle \( D \): 4.5,6,7.5. 4.5/4.5=1? No, wait, maybe \( T \) is Triangle \( A \) (4,5,6). Then Triangle \( D \): 4.5,6,7.5. 4.5/4=1.125, 6/5=1.2, 7.5/6=1.25. Not consistent.

Wait, maybe \( T \) is 5-6-4? No, let's think again. The key is that a scaled copy has all sides multiplied by the same scale factor. So for each triangle, check if the ratios of corresponding sides are equal.

Let's take Triangle \( B \) (3,4,5) and check others:

• Triangle \( D \): 4.5, 6, 7.5. Check ratios: 4.5/3 = 1.5, 6/4 = 1.5, 7.5/5 = 1.5. So scale factor 1.5.

• Triangle \( E \): 6,8,10. Ratios: 6/3=2, 8/4=2, 10/5=2. Scale factor 2.

• Triangle \( A \): 4,5,6. Ratios: 4/3 ≈1.333, 5/4=1.25, 6/5=1.2. Not equal. So not a scaled copy.

• Triangle \( C \): 4,5,6.4. Ratios: 4/3≈1.333, 5/4=1.25, 6.4/5=1.28. Not equal.

• Triangle \( F \): 6,7,8. Ratios: 6/3=2, 7/4=1.75, 8/5=1.6. Not equal.

So if \( T \) is Triangle \( B \) (3,4,5), then:

• Triangle \( A \): not a scaled copy (blank)
• Triangle \( B \): scale factor 1 (but maybe \( T \) is another)

Wait, maybe \( T \) is Triangle \( A \) (4,5,6). Let's check:

• Triangle \( D \): 4.5,6,7.5. Ratios: 4.5/4=1.125, 6/5=1.2, 7.5/6=1.25. Not equal.

• Triangle \( E \): 6,8,10. Ratios: 6/4=1.5, 8/5=1.6, 10/6≈1.666. Not equal.

Alternatively, maybe \( T \) is Triangle \( E \) (6,8,10). Then:

• Triangle \( B \): 3,4,5. Ratios: 3/6=0.5, 4/8=0.5, 5/10=0.5. Scale factor 0.5.

• Triangle \( D \): 4.5,6,7.5. Ratios: 4.5/6=0.75, 6/8=0.75, 7.5/10=0.75. Scale factor 0.75 (or 3/4).

• Triangle \( A \): 4,5,6. Ratios: 4/6≈0.666, 5/8=0.625, 6/10=0.6. Not equal.

• Triangle \( C \): 4,5,6.4. Ratios: 4/6≈0.666, 5/8=0.625, 6.4/10=0.64. Not equal.

• Triangle \( F \): 6,7,8. Ratios: 6/6=1, 7/8=0.875, 8/10=0.8. Not equal.

This is getting confusing. Let's look at the numbers again. The problem says "For each scaled copy, write the scale factor that takes triangle \( T \) to that triangle." So we need to know what triangle \( T \) is. Maybe \( T \) is the smallest one, Triangle \( B \) (3,4,5). Let's proceed with that.

So:

• Triangle \( A \): sides 4,5,6. Check if 3k=4, 4k=5, 5k=6? No, k would be 4/3, 5/4, 6/5, which are different. So not a scaled copy (blank).

• Triangle \( B \): same as \( T \), scale factor 1.

• Triangle \( C \): sides 4,5,6.4. 3k=4 → k=4/3; 4k=5 → k=5/4; 5k=6.4 → k=1.28. Not same. Blank.

• Triangle \( D \): sides 4.5,6,7.5. 3k=4.5 → k=1.5; 4k=6 → k=1.5; 5k=7.5 → k=1.5. So scale factor 1.5.

• Triangle \( E \): sides 6,8,10. 3k=6 → k=2; 4k=8 → k=2; 5k=10 → k=2. Scale factor 2.

• Triangle \( F \): sides 6,7,8. 3k=6 → k=2; 4k=7 → k=1.75; 5k=8 → k=1.6. Not same. Blank.

So the scale factors would be:

• Triangle \( A \): (blank)
• Triangle \( B \): 1
• Triangle \( C \): (blank)
• Triangle \( D \): 1.5
• Triangle \( E \): 2
• Triangle \( F \): (blank)

But maybe \( T \) is Triangle \( A \). Let's check:

Triangle \( A \): 4,5,6.

• Triangle \( D \): 4.5,6,7.5. 4k=4.5 → k=1.125; 5k=6 → k=1.2; 6k=7.5 → k=1.25. Not same.

• Triangle \( E \): 6,8,10. 4k=6 → k=1.5; 5k=8 → k=1.6; 6k=10 → k≈1.666. Not same.

Alternatively, maybe \( T \) is Triangle \( D \). No, that's more complicated.

Given the common Pythagorean triples, 3-4-5 (Triangle \( B \)) and its multiples (6-8-10 (Triangle \( E \)), 4.5-6-7.5 (Triangle \( D \), which is 3-4-5 scaled by 1.5)), these are scaled copies. So assuming \( T \) is 3-4-5 (Triangle \( B \)):

• Triangle \( A \): Not a scaled copy (blank)
• Triangle \( B \): Scale factor 1
• Triangle \( C \): Not a scaled copy (blank)
• Triangle \( D \): Scale factor 1.5
• Triangle \( E \): Scale factor 2
• Triangle \( F \): Not a scaled copy (blank)

So filling the table:

Triangle	Scale Factor
B	1
C	(blank)
D	1.5
E	2
F	(blank)

But the problem says "write the scale factor that takes triangle \( T \) to that triangle". So if \( T \) is, say, Triangle \( B \), then:

• For \( D \): scale factor 1.5 (since 31.5=4.5, 41.5=6, 5*1.5=7.5)
• For \( E \): scale factor 2 (32=6, 42=8, 5*2=10)
• For \( B \): scale factor 1
• Others: blank

So the answer depends on \( T \), but likely \( T \) is 3-4-5 (Triangle \( B \)).