Question

given △abc with coordinates: a(1,2), b(4,6), c(7,2) and △def, with coordinates: d(-1,-2), e(2,1), f(5,-2), use the distance formula to determine if the triangles are similar.
the triangles are similar because all corresponding sides are equal in length.
the triangles are similar because corresponding side lengths are proportional.
the triangles are not similar because the corresponding sides do not have the same slopes.
the triangles are not similar because the corresponding side lengths are not proportional.

Explanation:

Step1: Calculate side lengths of △ABC

For \( A(1,2) \), \( B(4,6) \), \( C(7,2) \):
- \( AB \): \( \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
- \( BC \): \( \sqrt{(7 - 4)^2 + (2 - 6)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
- \( AC \): \( \sqrt{(7 - 1)^2 + (2 - 2)^2} = \sqrt{36 + 0} = 6 \)

Step2: Calculate side lengths of △DEF

For \( D(-1,-2) \), \( E(2,1) \), \( F(5,-2) \):
- \( DE \): \( \sqrt{(2 - (-1))^2 + (1 - (-2))^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \)
- \( EF \): \( \sqrt{(5 - 2)^2 + (-2 - 1)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \)
- \( DF \): \( \sqrt{(5 - (-1))^2 + (-2 - (-2))^2} = \sqrt{36 + 0} = 6 \)

Step3: Check proportionality

- \( \frac{AB}{DE} = \frac{5}{3\sqrt{2}} \), \( \frac{BC}{EF} = \frac{5}{3\sqrt{2}} \), \( \frac{AC}{DF} = \frac{6}{6} = 1 \)
Wait, correction: Wait, AC is 6 (from x=1 to x=7, y same), DF is 6 (x=-1 to x=5, y same). AB: distance between (1,2) and (4,6): \( \sqrt{3^2 + 4^2}=5 \). DE: (-1,-2) to (2,1): \( \sqrt{3^2 + 3^2}=3\sqrt{2} \). BC: (4,6) to (7,2): \( \sqrt{3^2 + (-4)^2}=5 \). EF: (2,1) to (5,-2): \( \sqrt{3^2 + (-3)^2}=3\sqrt{2} \). So ratios: \( \frac{AB}{DE}=\frac{5}{3\sqrt{2}} \), \( \frac{BC}{EF}=\frac{5}{3\sqrt{2}} \), \( \frac{AC}{DF}=\frac{6}{6}=1 \). Wait, no, AC is 6 (length), DF is 6 (length). Wait, AB and BC are 5, DE and EF are \( 3\sqrt{2} \approx 4.24 \), AC and DF are 6. Wait, maybe I miscalculated. Wait, AC: from (1,2) to (7,2): horizontal distance, 6 units. DF: from (-1,-2) to (5,-2): horizontal distance, 6 units. AB: (1,2) to (4,6): x difference 3, y difference 4, so 5. DE: (-1,-2) to (2,1): x difference 3, y difference 3, so \( 3\sqrt{2} \). BC: (4,6) to (7,2): x difference 3, y difference -4, so 5. EF: (2,1) to (5,-2): x difference 3, y difference -3, so \( 3\sqrt{2} \). So the sides: AB=5, BC=5, AC=6; DE=3√2, EF=3√2, DF=6. Now check ratios: \( \frac{AB}{DE} = \frac{5}{3\sqrt{2}} \), \( \frac{BC}{EF} = \frac{5}{3\sqrt{2}} \), \( \frac{AC}{DF} = \frac{6}{6} = 1 \). Wait, that's not consistent. Wait, no, maybe the triangles are isoceles? Wait, △ABC: AB=5, BC=5, AC=6 (isoceles). △DEF: DE=3√2, EF=3√2, DF=6 (isoceles). Now check if the ratios of corresponding sides are equal. Let's pair AB with DE, BC with EF, AC with DF. \( \frac{AB}{DE} = \frac{5}{3\sqrt{2}} \approx 1.178 \), \( \frac{BC}{EF} = \frac{5}{3\sqrt{2}} \approx 1.178 \), \( \frac{AC}{DF} = 1 \). Wait, that's not equal. Wait, maybe I paired wrong. Wait, maybe AC corresponds to DF, AB to DE, BC to EF. Wait, no, maybe the triangles have angles equal? Wait, AC is horizontal (y=2), DF is horizontal (y=-2). AB: slope (6-2)/(4-1)=4/3, DE: slope (1 - (-2))/(2 - (-1))=3/3=1. BC: slope (2-6)/(7-4)=-4/3, EF: slope (-2 - 1)/(5 - 2)=-3/3=-1. So angles at A and D: AB has slope 4/3, AC is horizontal (slope 0), so angle at A: arctan(4/3). DE has slope 1, DF is horizontal (slope 0), so angle at D: arctan(1). These angles are not equal, so triangles are not similar? Wait, no, wait the third option says "The triangles are similar because corresponding side lengths are proportional." Wait, maybe my calculation is wrong. Wait, let's recalculate AB: distance between (1,2) and (4,6): \( \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = 5 \). DE: (-1,-2) to (2,1): \( \sqrt{(2+1)^2 + (1+2)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \approx 4.2426 \). AC: (1,2) to (7,2): 6. DF: (-1,-2) to (5,-2): 6. BC: (4,6) to (7,2): \( \sqrt{(7-4)^2 + (2-6)^2} = \sqrt{9 + 16} = 5 \). EF: (2,1) to (5,-2): \( \sqrt{(5-2)^2 + (-2-1)^2} = \sqrt{9 + 9} = 3\sqrt{2} \approx 4.2426 \). Now, let's check the ratios: AB/DE = 5 / (3√2) ≈ 5 / 4.2426 ≈ 1.1785. BC/EF = 5 / (3√2) ≈ 1.1785. AC/DF = 6 / 6 = 1. Wait, these ratios are not equal, so sides are not proportional. But the third option says they are similar because sides are proportional. Wait, maybe I mixed up the triangles. Wait, the first triangle is ABC with A(1,2), B(4,6), C(7,2). The second is DEF with D(-1,-2), E(2,1), F(5,-2). Wait, maybe the sides are AB=5, BC=5, AC=6; DE=3√2, EF=3√2, DF=6. Wait, 5 / 3√2 = 5√2 / 6 ≈ 1.178, and 6 / 6 = 1. So not proportional. But the third option is "The triangles are similar because corresponding side lengths are proportional." Wait, maybe I made a mistake. Wait, let's check the distance formula again. For AB: (1,2) to (4,6): Δx=3, Δy=4, so distance 5. Correct. DE: (-1,-2) to (2,1): Δx=3, Δy=3, distance 3√2. Correct. AC: (1,2) to (7,2): Δx=6, Δy=0, distance 6. Correct. DF: (-1,-2) to (5,-2): Δx=6, Δy=0, distance 6. Correct. BC: (4,6) to (7,2): Δx=3, Δy=-4, distance 5. Correct. EF: (2,1) to (5,-2): Δx=3, Δy=-3, distance 3√2. Correct. So the sides of ABC: 5, 5, 6. Sides of DEF: 3√2, 3√2, 6. Now, let's see if 5/3√2 = 5/3√2 = 6/6? No, 6/6=1, 5/3√2≈1.178. So the ratios are not equal. Therefore, triangles are not similar because corresponding side lengths are not proportional. Wait, but the options: first option: "The triangles are similar because all corresponding sides are equal in length." No, sides are not equal. Second: "The triangles are similar because corresponding side lengths are proportional." But they are not. Third: "The triangles are not similar because the corresponding sides do not have the same slopes." No, slopes are about angles, but similarity is about sides or angles. Fourth: "The triangles are not similar because the corresponding side lengths are not proportional." Yes, that's correct. Wait, but the third option in the choices (the radio buttons) – let's check the options again:

Options:

1. The triangles are similar because all corresponding sides are equal in length. (No, sides are not equal)

2. The triangles are similar because corresponding side lengths are proportional. (No, ratios not equal)

3. The triangles are not similar because the corresponding sides do not have the same slopes. (Slopes relate to angles, but similarity is about sides or angles; but actually, the side length ratios are not proportional, so this is not the reason)

4. The triangles are not similar because the corresponding side lengths are not proportional. (Yes, this is correct)

Wait, maybe I messed up the option numbering. Let's re-express the options as per the image:

First option: "The triangles are similar because all corresponding sides are equal in length."

Second: "The triangles are similar because corresponding side lengths are proportional."

Third: "The triangles are not similar because the corresponding sides do not have the same slopes."

Fourth: "The triangles are not similar because the corresponding side lengths are not proportional."

So the correct answer is the fourth option: "The triangles are not similar because the corresponding side lengths are not proportional."

Answer:

The triangles are not similar because the corresponding side lengths are not proportional. (The fourth option, assuming the options are ordered as: 1. The triangles are similar because all corresponding sides are equal in length. 2. The triangles are similar because corresponding side lengths are proportional. 3. The triangles are not similar because the corresponding sides do not have the same slopes. 4. The triangles are not similar because the corresponding side lengths are not proportional.)