Question

problem 4 select two triangles whose lengths can be represented by the equivalent ratios (\frac{4}{8} = \frac{5}{10} = \frac{3}{6}).

Explanation:

Step1: Analyze Triangle T (sides 3, 4, 5)

Check the ratios of sides for each triangle. For similar triangles, the ratios of corresponding sides should be equal. Let's list the sides of each triangle:
- Triangle T: 3, 4, 5 (ratio of sides: \( \frac{3}{4} = 0.75 \), \( \frac{3}{5} = 0.6 \), \( \frac{4}{5} = 0.8 \))
- Triangle B: 3, 4, 5 (same as T, so ratios are equal)
- Triangle A: 4, 5, 6 (ratios: \( \frac{4}{5}=0.8 \), \( \frac{4}{6}\approx0.666 \), \( \frac{5}{6}\approx0.833 \))
- Triangle C: 4, 5, 6.4 (ratios: \( \frac{4}{5}=0.8 \), \( \frac{4}{6.4}=0.625 \), \( \frac{5}{6.4}\approx0.781 \))
- Triangle D: 4.5, 6, 7.5 (simplify by dividing by 1.5: 3, 4, 5. So ratios are \( \frac{4.5}{6}=0.75 \), \( \frac{4.5}{7.5}=0.6 \), \( \frac{6}{7.5}=0.8 \), same as T)
- Triangle E: 6, 8, 10 (divide by 2: 3, 4, 5. Ratios: \( \frac{6}{8}=0.75 \), \( \frac{6}{10}=0.6 \), \( \frac{8}{10}=0.8 \), same as T)
- Triangle F: 6, 7, 8 (ratios: \( \frac{6}{7}\approx0.857 \), \( \frac{6}{8}=0.75 \), \( \frac{7}{8}=0.875 \))

Step2: Identify Similar Triangles

Triangles with side ratios 3:4:5 are similar. Let's check:
- Triangle T (3,4,5), Triangle B (3,4,5) – same sides, so similar.
- Triangle T, Triangle D (4.5,6,7.5: 3,4,5 when divided by 1.5), Triangle E (6,8,10: 3,4,5 when divided by 2) are similar. Also, let's check the given equivalent ratios \( \frac{4}{8}=\frac{5}{10}=\frac{3}{6} \) (simplify: \( \frac{1}{2}=\frac{1}{2}=\frac{1}{2} \)). Wait, maybe I misread. Wait, the problem says "equivalent ratios \( \frac{4}{8}=\frac{5}{10}=\frac{3}{6} \)". Let's check which triangles have sides in ratio 3:4:5 or 3:4:5 scaled. Wait, \( \frac{3}{6}=\frac{1}{2} \), \( \frac{4}{8}=\frac{1}{2} \), \( \frac{5}{10}=\frac{1}{2} \). So triangles with sides 3,4,5 and 6,8,10 (since 3*2=6, 4*2=8, 5*2=10) or 4.5,6,7.5 (3*1.5=4.5, 4*1.5=6, 5*1.5=7.5) or 1.5,2,2.5 (but not here). Wait, looking at the triangles:
- Triangle T: 3,4,5
- Triangle E: 6,8,10 (3*2, 4*2, 5*2) – so \( \frac{3}{6}=\frac{4}{8}=\frac{5}{10}=\frac{1}{2} \), which matches the given equivalent ratios.
- Also, Triangle D: 4.5,6,7.5 (3*1.5, 4*1.5, 5*1.5) – \( \frac{3}{4.5}=\frac{4}{6}=\frac{5}{7.5}=\frac{2}{3} \), not the given ratio. Wait, the given ratio is \( \frac{4}{8}=\frac{5}{10}=\frac{3}{6} \) (all equal to 1/2). So sides 3,4,5 and 6,8,10 (since 3/6=1/2, 4/8=1/2, 5/10=1/2). So Triangle T (3,4,5) and Triangle E (6,8,10) have sides in ratio 1/2. Also, check Triangle A: 4,5,6 – no. Triangle C: 4,5,6.4 – no. Triangle B: 3,4,5 (same as T, ratio 1/1). Triangle F: 6,7,8 – no. Wait, maybe the problem is to find two triangles with sides in the ratio given (4/8=5/10=3/6). So 3,4,5 and 6,8,10 (Triangle E) or 3,4,5 and 6,8,10. Let's confirm the sides:
- Triangle T: 3,4,5
- Triangle E: 6,8,10 (3*2, 4*2, 5*2) – so the ratios of corresponding sides are 3/6=4/8=5/10=1/2, which are equivalent.
- Also, Triangle D: 4.5,6,7.5 – 3*1.5, 4*1.5, 5*1.5 – ratio 3/4.5=4/6=5/7.5=2/3, not 1/2.
- Triangle B: 3,4,5 (same as T, ratio 1)
- So the two triangles could be T and E, or T and B (but B is same as T, maybe E is scaled by 2). Let's check the given equivalent ratios: 4/8=1/2, 5/10=1/2, 3/6=1/2. So sides 3,4,5 (T) and 6,8,10 (E) have sides 3,4,5 and 6,8,10, so 3/6=4/8=5/10=1/2. So those two triangles (T and E) or maybe another pair. Wait, also Triangle A: 4,5,6 – no. Triangle C: 4,5,6.4 – 4/8? No, 4/6.4=0.625, 5/10=0.5, not equal. Wait, maybe I made a mistake. Let's list all triangles with their side lengths (assuming the labels and sides):
- T: 3, 4, 5
- B: 3, 4, 5 (same as T)
- A: 4, 5, 6
- C: 4, 5, 6.4
- D: 4.5, 6, 7.5 (4.5, 6, 7.5)
- E: 6, 8, 10 (6, 8, 10)
- F: 6, 7, 8 (6, 7, 8)

Now, check the ratio \( \frac{4}{8} = \frac{5}{10} = \frac{3}{6} \). Simplify each: \( \frac{4}{8} = \frac{1}{2} \), \( \frac{5}{10} = \frac{1}{2} \), \( \frac{3}{6} = \frac{1}{2} \). So we need two triangles where the sides are in the ratio 3:4:5 and 6:8:10 (since 3*2=6, 4*2=8, 5*2=10), so the sides are scaled by 2. So Triangle T (3,4,5) and Triangle E (6,8,10) have sides in ratio 1/2, which matches the given equivalent ratios. Alternatively, Triangle B (3,4,5) and Triangle E (6,8,10) also work. Or Triangle T and Triangle D? Wait, Triangle D: 4.5,6,7.5. 3/4.5=2/3, 4/6=2/3, 5/7.5=2/3. So that's a ratio of 2/3, not 1/2. So the correct pair is T (3,4,5) and E (6,8,10), or B (3,4,5) and E (6,8,10), or T and B (but B is same as T, maybe E is the scaled one). Let's confirm with the given ratios: 4/8=1/2, 5/10=1/2, 3/6=1/2. So the sides 3,4,5 (T) and 6,8,10 (E) have 3/6=1/2, 4/8=1/2, 5/10=1/2, so they are similar with scale factor 2.

Answer:

Two triangles are Triangle T (sides 3, 4, 5) and Triangle E (sides 6, 8, 10) (or Triangle B and Triangle E, or Triangle T and Triangle B, but the most probable with the given ratio \( \frac{4}{8}=\frac{5}{10}=\frac{3}{6} \) is T and E or B and E).