Question

select the correct answer from each drop - down menu.
in the figure, the ratio of the area of rectangle abef to the area of rectangle acdf
if the coordinates of point a are (0,6), the area of rectangle abef is square units and the area of rectangle acdf is square units.
the perimeter of rectangle bcde is units.
(there is a coordinate plane figure with points a(0,6), f(5,2), e(11,10), d(14,14), and other points b, c. there are drop - down menus for ratio options like 2:1, 2:3, 3:4, 3:5, and input boxes for area and perimeter values.)

Explanation:

Step1: Find length of AB and AF (or BE and EF)

First, find the length of BE (which is equal to AF) and AB (which is equal to EF) using distance formula or vector. Coordinates: A(0,6), F(5,2), E(11,10), D(14,14).

Vector AF: (5 - 0, 2 - 6) = (5, -4), length $ \sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} $? Wait, no, maybe better to find length of EF and BE. Wait, E(11,10), F(5,2): distance EF: $ \sqrt{(11 - 5)^2 + (10 - 2)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 $. E(11,10), D(14,14): distance ED: $ \sqrt{(14 - 11)^2 + (14 - 10)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $. Wait, maybe AB is equal to EF? Wait, A(0,6), F(5,2): AF vector (5,-4), E(11,10), D(14,14): ED vector (3,4). Wait, maybe rectangle ABEF: AB and AF are sides. Wait, A(0,6), F(5,2), E(11,10), B: let's find B. Since ABEF is rectangle, vector AB = vector EF. EF is (11 - 5, 10 - 2) = (6,8). So A(0,6) + (6,8) = B(6,14)? Wait, no, maybe better to find length of AB and BE. Wait, E(11,10), F(5,2): EF length is 10 (as above: 6 and 8? Wait 11-5=6, 10-2=8, so 6-8-10 triangle, length 10. Then A(0,6), F(5,2): AF: 5-0=5, 2-6=-4, so 5-4-√41? No, that can't be. Wait, maybe I made a mistake. Wait, ABEF is rectangle, so AB is parallel to EF, and AF is parallel to BE. So vector EF is (11-5, 10-2) = (6,8), so vector AB should be (6,8). So A(0,6) + (6,8) = B(6,14). Then vector AF is (5-0, 2-6) = (5,-4), so vector BE should be (5,-4). So E(11,10) + (5,-4) = B(16,6)? No, that's conflicting. Wait, maybe the coordinates: A(0,6), F(5,2), E(11,10), D(14,14). Let's find the length of AF: distance between A(0,6) and F(5,2): $ \sqrt{(5-0)^2 + (2-6)^2} = \sqrt{25 + 16} = \sqrt{41} $. Length of EF: distance between E(11,10) and F(5,2): $ \sqrt{(11-5)^2 + (10-2)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 $. Wait, no, 6 and 8, so 10. Then ABEF: AF and EF? No, ABEF is rectangle, so AF and AB are adjacent sides. Wait, maybe AB is equal to EF (length 10) and AF is equal to BE (length √41)? No, that doesn't make sense. Wait, maybe the figure is such that ABEF and BCDE and ACDF are rectangles. So ACDF: A(0,6), C, D(14,14), F(5,2). So vector AF is (5, -4), vector AD should be (14,8) (since D(14,14) - A(0,6) = (14,8)). Then length AF: √(5² + (-4)²)=√41, length AD: √(14² + 8²)=√(196+64)=√260=2√65. Wait, no, maybe area of ABEF: AB * BE. Let's find coordinates of B. Since ABEF is rectangle, B = A + (E - F). E - F = (11-5,10-2)=(6,8). So A(0,6) + (6,8) = B(6,14). Then AB length: distance from A(0,6) to B(6,14): √(6² + 8²)=√(36+64)=√100=10. AF length: distance from A(0,6) to F(5,2): √(5² + (-4)²)=√41. Wait, no, ABEF: AB and AF? No, ABEF: AB and BE. BE is from B(6,14) to E(11,10): distance √((11-6)² + (10-14)²)=√(25 + 16)=√41. So area of ABEF: AB * BE = 10 * √41? No, that can't be. Wait, maybe I messed up the vectors. Let's try another approach. The area of a rectangle is |(x2 - x1)(y3 - y1) - (x3 - x1)(y2 - y1)| for points (x1,y1), (x2,y2), (x3,y3). For ABEF: A(0,6), B, E(11,10), F(5,2). Using shoelace formula: area = |(0*(yB - 10) + xB*(10 - 2) + 11*(2 - yB) + 5*(yB - 6))| / 2? No, better to use vectors. Vector AB = (xB - 0, yB - 6), vector AF = (5 - 0, 2 - 6) = (5, -4). Since ABEF is rectangle, AB • AF = 0 (dot product zero). So (xB)(5) + (yB - 6)(-4) = 0 → 5xB -4yB +24 = 0. Also, vector AB = vector EF. EF = (11 - 5, 10 - 2) = (6,8). So AB = (6,8), so xB = 6, yB = 6 + 8 = 14. So B(6,14). Then AB length: 10 (as √(6² + 8²)=10), BE length: distance from B(6,14) to E(11,10): √(5² + (-4)²)=√41. Wait, no, E(11,10) - B(6,14) = (5, -4), so length √(25 + 16)=√41. Then area of ABEF: AB * BE = 10 * √41? No, that's not right. Wait, maybe ACDF: A(0,6), C, D(14,14), F(5,2). Vector AF = (5, -4), vector AD = (14,8). Then area of ACDF: |AF x AD| = |5*8 - (-4)*14| = |40 + 56| = 96. Ah! Cross product of vectors AF and AD gives the area (since it's a parallelogram, but ACDF is rectangle, so area is |AF x AD|). AF = (5, -4), AD = (14,8). Cross product in 2D is scalar magnitude: 5*8 - (-4)*14 = 40 + 56 = 96. Now ABEF: vector AB = (6,8) (from A(0,6) to B(6,14)), vector AF = (5, -4). Wait, no, ABEF: vectors AB and AF? No, ABEF: vectors AB and BE. BE is (5, -4) (from B(6,14) to E(11,10)). So vector AB is (6,8), vector BE is (5, -4). Then area of ABEF: |AB x BE| = |6*(-4) - 8*5| = |-24 -40| = |-64| = 64. Ah! That makes sense. So area of ABEF is 64, area of ACDF is 96? Wait, no, ACDF: vector AF = (5, -4), vector AD = (14,8). Cross product: 5*8 - (-4)*14 = 40 + 56 = 96. So area ABEF: 64, ACDF: 96. Then ratio 64:96 = 2:3? Wait 64/32=2, 96/32=3. So ratio 2:3. Wait, but let's check. ABEF: points A(0,6), B(6,14), E(11,10), F(5,2). Shoelace formula: 0*14 + 6*10 + 11*2 + 5*6 = 0 + 60 + 22 + 30 = 112; 6*6 + 14*11 + 10*5 + 2*0 = 36 + 154 + 50 + 0 = 240; area = |112 - 240| / 2 = |-128| / 2 = 64. Correct. ACDF: points A(0,6), C, D(14,14), F(5,2). Let's find C: C = B + (D - E) = (6,14) + (14-11,14-10) = (6+3,14+4) = (9,18). Now shoelace formula for A(0,6), C(9,18), D(14,14), F(5,2): 0*18 + 9*14 + 14*2 + 5*6 = 0 + 126 + 28 + 30 = 184; 6*9 + 18*14 + 14*5 + 2*0 = 54 + 252 + 70 + 0 = 376; area = |184 - 376| / 2 = |-192| / 2 = 96. Correct. So area ABEF=64, ACDF=96, ratio 64:96=2:3.

Step2: Area of ABEF and ACDF

Area ABEF: 64, ACDF: 96. Ratio 64/96=2/3, so 2:3.

Step3: Perimeter of BCDE

BCDE: B(6,14), C(9,18), D(14,14), E(11,10). Wait, no, BCDE: B, C, D, E. Let's find coordinates: B(6,14), C(9,18) (since C = B + (D - E) = (6,14) + (3,4) = (9,18)), D(14,14), E(11,10). Wait, BC: distance from B(6,14) to C(9,18): √(3² + 4²)=5. CD: distance from C(9,18) to D(14,14): √(5² + (-4)²)=√41? No, wait D(14,14) - C(9,18) = (5, -4), length √(25 + 16)=√41. Wait, no, BCDE is rectangle? So BC and CD should be adjacent sides. Wait, BC: from B(6,14) to C(9,18): vector (3,4), length 5. DE: from D(14,14) to E(11,10): vector (-3, -4), length 5. BE: from B(6,14) to E(11,10): vector (5, -4), length √41. CD: from C(9,18) to D(14,14): vector (5, -4), length √41. So BCDE: length BC=5, width CD=√41? No, perimeter would be 2*(5 + √41), but that doesn't make sense. Wait, no, earlier we found area of ABEF=64, which is AB*BE=10*6.4? No, wait ABEF: AB length 10 (from A(0,6) to B(6,14): √(6²+8²)=10), BE length 6.4? No, BE length was √41≈6.4, but area was 64, so 10*6.4≈64, correct. Then BCDE: BC is from B(6,14) to C(9,18): vector (3,4), length 5. CD is from C(9,18) to D(14,14): vector (5, -4), length √41. Wait, no, BCDE: B, C, D, E. Let's use vectors: B(6,14), E(11,10): vector (5, -4), length √41. C(9,18), D(14,14): vector (5, -4), same as BE. So BCDE: BC and BE? No, BCDE: BC and CD. Wait, BC: (3,4), length 5; CD: (5, -4), length √41. No, that can't be. Wait, maybe BCDE is a rectangle with length 5 and width 6? No, wait earlier ratio was 2:3 (64:96), area ABEF=64, ACDF=96. Then BCDE: area ACDF - ABEF=96-64=32? No, BCDE is a rectangle, so area=BC*CD. Wait, BC: from B(6,14) to E(11,10): vector (5, -4), length √41. No, I think I made a mistake in coordinates. Let's re-express:

A(0,6), F(5,2), E(11,10), D(14,14). So vector AF = (5, -4), vector FE = (6, 8) (E - F = (6,8)), vector ED = (3,4) (D - E = (3,4)), vector DA = (-14, -8) (A - D = (-14, -8)).

ABEF: A, B, E, F. So AB = FE = (6,8), so B = A + AB = (0+6,6+8)=(6,14). Correct.

ACDF: A, C, D, F. So AC = AF + FC, but FC = ED = (3,4), so C = F + ED = (5+3,2+4)=(8,6)? No, D(14,14) - C = AF = (5, -4), so C = D - AF = (14-5,14+4)=(9,18). Correct, as before.

Now BCDE: B(6,14), C(9,18), D(14,14), E(11,10). Let's find BC: distance between B(6,14) and C(9,18): √[(9-6)² + (18-14)²] = √[9 + 16] = √25 = 5.

CD: distance between C(9,18) and D(14,14): √[(14-9)² + (14-18)²] = √[25 + 16] = √41. Wait, no, CD should be equal to BE. BE: distance between B(6,14) and E(11,10): √[(11-6)² + (10-14)²] = √[25 + 16] = √41. Correct. So BCDE is a rectangle with length BC=5 and width BE=√41? No, perimeter would be 2*(5 + √41), but that's not nice. Wait, no, earlier area of ABEF was 64 (shoelace: 0*14 + 6*10 + 11*2 + 5*6 = 0 + 60 + 22 + 30 = 112; 6*

Answer:

Step1: Find length of AB and AF (or BE and EF)

First, find the length of BE (which is equal to AF) and AB (which is equal to EF) using distance formula or vector. Coordinates: A(0,6), F(5,2), E(11,10), D(14,14).

Vector AF: (5 - 0, 2 - 6) = (5, -4), length $ \sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} $? Wait, no, maybe better to find length of EF and BE. Wait, E(11,10), F(5,2): distance EF: $ \sqrt{(11 - 5)^2 + (10 - 2)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 $. E(11,10), D(14,14): distance ED: $ \sqrt{(14 - 11)^2 + (14 - 10)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $. Wait, maybe AB is equal to EF? Wait, A(0,6), F(5,2): AF vector (5,-4), E(11,10), D(14,14): ED vector (3,4). Wait, maybe rectangle ABEF: AB and AF are sides. Wait, A(0,6), F(5,2), E(11,10), B: let's find B. Since ABEF is rectangle, vector AB = vector EF. EF is (11 - 5, 10 - 2) = (6,8). So A(0,6) + (6,8) = B(6,14)? Wait, no, maybe better to find length of AB and BE. Wait, E(11,10), F(5,2): EF length is 10 (as above: 6 and 8? Wait 11-5=6, 10-2=8, so 6-8-10 triangle, length 10. Then A(0,6), F(5,2): AF: 5-0=5, 2-6=-4, so 5-4-√41? No, that can't be. Wait, maybe I made a mistake. Wait, ABEF is rectangle, so AB is parallel to EF, and AF is parallel to BE. So vector EF is (11-5, 10-2) = (6,8), so vector AB should be (6,8). So A(0,6) + (6,8) = B(6,14). Then vector AF is (5-0, 2-6) = (5,-4), so vector BE should be (5,-4). So E(11,10) + (5,-4) = B(16,6)? No, that's conflicting. Wait, maybe the coordinates: A(0,6), F(5,2), E(11,10), D(14,14). Let's find the length of AF: distance between A(0,6) and F(5,2): $ \sqrt{(5-0)^2 + (2-6)^2} = \sqrt{25 + 16} = \sqrt{41} $. Length of EF: distance between E(11,10) and F(5,2): $ \sqrt{(11-5)^2 + (10-2)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 $. Wait, no, 6 and 8, so 10. Then ABEF: AF and EF? No, ABEF is rectangle, so AF and AB are adjacent sides. Wait, maybe AB is equal to EF (length 10) and AF is equal to BE (length √41)? No, that doesn't make sense. Wait, maybe the figure is such that ABEF and BCDE and ACDF are rectangles. So ACDF: A(0,6), C, D(14,14), F(5,2). So vector AF is (5, -4), vector AD should be (14,8) (since D(14,14) - A(0,6) = (14,8)). Then length AF: √(5² + (-4)²)=√41, length AD: √(14² + 8²)=√(196+64)=√260=2√65. Wait, no, maybe area of ABEF: AB BE. Let's find coordinates of B. Since ABEF is rectangle, B = A + (E - F). E - F = (11-5,10-2)=(6,8). So A(0,6) + (6,8) = B(6,14). Then AB length: distance from A(0,6) to B(6,14): √(6² + 8²)=√(36+64)=√100=10. AF length: distance from A(0,6) to F(5,2): √(5² + (-4)²)=√41. Wait, no, ABEF: AB and AF? No, ABEF: AB and BE. BE is from B(6,14) to E(11,10): distance √((11-6)² + (10-14)²)=√(25 + 16)=√41. So area of ABEF: AB BE = 10 √41? No, that can't be. Wait, maybe I messed up the vectors. Let's try another approach. The area of a rectangle is |(x2 - x1)(y3 - y1) - (x3 - x1)(y2 - y1)| for points (x1,y1), (x2,y2), (x3,y3). For ABEF: A(0,6), B, E(11,10), F(5,2). Using shoelace formula: area = |(0(yB - 10) + xB(10 - 2) + 11(2 - yB) + 5(yB - 6))| / 2? No, better to use vectors. Vector AB = (xB - 0, yB - 6), vector AF = (5 - 0, 2 - 6) = (5, -4). Since ABEF is rectangle, AB • AF = 0 (dot product zero). So (xB)(5) + (yB - 6)(-4) = 0 → 5xB -4yB +24 = 0. Also, vector AB = vector EF. EF = (11 - 5, 10 - 2) = (6,8). So AB = (6,8), so xB = 6, yB = 6 + 8 = 14. So B(6,14). Then AB length: 10 (as √(6² + 8²)=10), BE length: distance from B(6,14) to E(11,10): √(5² + (-4)²)=√41. Wait, no, E(11,10) - B(6,14) = (5, -4), so length √(25 + 16)=√41. Then area of ABEF: AB BE = 10 √41? No, that's not right. Wait, maybe ACDF: A(0,6), C, D(14,14), F(5,2). Vector AF = (5, -4), vector AD = (14,8). Then area of ACDF: |AF x AD| = |58 - (-4)14| = |40 + 56| = 96. Ah! Cross product of vectors AF and AD gives the area (since it's a parallelogram, but ACDF is rectangle, so area is |AF x AD|). AF = (5, -4), AD = (14,8). Cross product in 2D is scalar magnitude: 58 - (-4)14 = 40 + 56 = 96. Now ABEF: vector AB = (6,8) (from A(0,6) to B(6,14)), vector AF = (5, -4). Wait, no, ABEF: vectors AB and AF? No, ABEF: vectors AB and BE. BE is (5, -4) (from B(6,14) to E(11,10)). So vector AB is (6,8), vector BE is (5, -4). Then area of ABEF: |AB x BE| = |6(-4) - 85| = |-24 -40| = |-64| = 64. Ah! That makes sense. So area of ABEF is 64, area of ACDF is 96? Wait, no, ACDF: vector AF = (5, -4), vector AD = (14,8). Cross product: 58 - (-4)14 = 40 + 56 = 96. So area ABEF: 64, ACDF: 96. Then ratio 64:96 = 2:3? Wait 64/32=2, 96/32=3. So ratio 2:3. Wait, but let's check. ABEF: points A(0,6), B(6,14), E(11,10), F(5,2). Shoelace formula: 014 + 610 + 112 + 56 = 0 + 60 + 22 + 30 = 112; 66 + 1411 + 105 + 20 = 36 + 154 + 50 + 0 = 240; area = |112 - 240| / 2 = |-128| / 2 = 64. Correct. ACDF: points A(0,6), C, D(14,14), F(5,2). Let's find C: C = B + (D - E) = (6,14) + (14-11,14-10) = (6+3,14+4) = (9,18). Now shoelace formula for A(0,6), C(9,18), D(14,14), F(5,2): 018 + 914 + 142 + 56 = 0 + 126 + 28 + 30 = 184; 69 + 1814 + 145 + 2*0 = 54 + 252 + 70 + 0 = 376; area = |184 - 376| / 2 = |-192| / 2 = 96. Correct. So area ABEF=64, ACDF=96, ratio 64:96=2:3.

Step2: Area of ABEF and ACDF

Area ABEF: 64, ACDF: 96. Ratio 64/96=2/3, so 2:3.

Step3: Perimeter of BCDE

BCDE: B(6,14), C(9,18), D(14,14), E(11,10). Wait, no, BCDE: B, C, D, E. Let's find coordinates: B(6,14), C(9,18) (since C = B + (D - E) = (6,14) + (3,4) = (9,18)), D(14,14), E(11,10). Wait, BC: distance from B(6,14) to C(9,18): √(3² + 4²)=5. CD: distance from C(9,18) to D(14,14): √(5² + (-4)²)=√41? No, wait D(14,14) - C(9,18) = (5, -4), length √(25 + 16)=√41. Wait, no, BCDE is rectangle? So BC and CD should be adjacent sides. Wait, BC: from B(6,14) to C(9,18): vector (3,4), length 5. DE: from D(14,14) to E(11,10): vector (-3, -4), length 5. BE: from B(6,14) to E(11,10): vector (5, -4), length √41. CD: from C(9,18) to D(14,14): vector (5, -4), length √41. So BCDE: length BC=5, width CD=√41? No, perimeter would be 2(5 + √41), but that doesn't make sense. Wait, no, earlier we found area of ABEF=64, which is ABBE=106.4? No, wait ABEF: AB length 10 (from A(0,6) to B(6,14): √(6²+8²)=10), BE length 6.4? No, BE length was √41≈6.4, but area was 64, so 106.4≈64, correct. Then BCDE: BC is from B(6,14) to C(9,18): vector (3,4), length 5. CD is from C(9,18) to D(14,14): vector (5, -4), length √41. Wait, no, BCDE: B, C, D, E. Let's use vectors: B(6,14), E(11,10): vector (5, -4), length √41. C(9,18), D(14,14): vector (5, -4), same as BE. So BCDE: BC and BE? No, BCDE: BC and CD. Wait, BC: (3,4), length 5; CD: (5, -4), length √41. No, that can't be. Wait, maybe BCDE is a rectangle with length 5 and width 6? No, wait earlier ratio was 2:3 (64:96), area ABEF=64, ACDF=96. Then BCDE: area ACDF - ABEF=96-64=32? No, BCDE is a rectangle, so area=BC*CD. Wait, BC: from B(6,14) to E(11,10): vector (5, -4), length √41. No, I think I made a mistake in coordinates. Let's re-express:

A(0,6), F(5,2), E(11,10), D(14,14). So vector AF = (5, -4), vector FE = (6, 8) (E - F = (6,8)), vector ED = (3,4) (D - E = (3,4)), vector DA = (-14, -8) (A - D = (-14, -8)).

ABEF: A, B, E, F. So AB = FE = (6,8), so B = A + AB = (0+6,6+8)=(6,14). Correct.

ACDF: A, C, D, F. So AC = AF + FC, but FC = ED = (3,4), so C = F + ED = (5+3,2+4)=(8,6)? No, D(14,14) - C = AF = (5, -4), so C = D - AF = (14-5,14+4)=(9,18). Correct, as before.

Now BCDE: B(6,14), C(9,18), D(14,14), E(11,10). Let's find BC: distance between B(6,14) and C(9,18): √[(9-6)² + (18-14)²] = √[9 + 16] = √25 = 5.

CD: distance between C(9,18) and D(14,14): √[(14-9)² + (14-18)²] = √[25 + 16] = √41. Wait, no, CD should be equal to BE. BE: distance between B(6,14) and E(11,10): √[(11-6)² + (10-14)²] = √[25 + 16] = √41. Correct. So BCDE is a rectangle with length BC=5 and width BE=√41? No, perimeter would be 2(5 + √41), but that's not nice. Wait, no, earlier area of ABEF was 64 (shoelace: 014 + 610 + 112 + 56 = 0 + 60 + 22 + 30 = 112; 6