画图解答---**Question 6:** **Question Stem:** 如图，四边形ABFD是平行四边形，四边形CDEF是正方形，四边形AGHF是长方形。已知AD=14 cm, BC=22 cm，阴影部分的面积是多少平方厘米？ **Translation of Question Stem:** As shown in the figure, quadrilateral ABFD is a parallelogram, quadrilateral CDEF is a square, and quadrilateral AGHF is a rectangle. Given AD=14 cm and BC=22 cm, what is the area of the shaded region in square centimeters? **Options:** (No options provided in the image) **Other Relevant Text/Calculations:** * "不会做题扫这里" (Scan here if you can't do the problem - refers to the QR code). * Handwritten notes below the diagram: "BE=14", "h=8", "}S阴" (implied calculation related to the shaded area S_yin). **Chart/Diagram Description:** * **Type:** Geometric figure. * **Main Elements:** * **Points:** Labeled points are A, B, C, D, E, F, G, H. * **Lines:** Straight lines connect the points forming quadrilaterals and triangles. Some lines are part of the outlines, others are internal diagonals or segments. * **Shapes:** * Quadrilateral ABFD is described as a parallelogram. * Quadrilateral CDEF is described as a square. * Quadrilateral AGHF is described as a rectangle. * Triangles: △ABE, △BCF, △BFD, △BDC, △ADE, △ADC, △CDE, △BDF, △BFG, △BGH, △AGH, △AHF, etc. * **Labels and Annotations:** * Length AD is labeled as 14 cm (indicated above AD). * Length BC is labeled as 22 cm (indicated below BC). * Length AE is labeled as 6. * Length ED is labeled as 8. (Note: AE + ED = 6 + 8 = 14 = AD, consistent with the given AD=14). * Side length of the square CDEF is labeled as 8 (indicated vertically along DE and horizontally along CD). This implies CD=DE=EF=FC=8. * Length of BF is labeled as 14 (indicated along BF). * Length of BF is labeled again as 14 (indicated along BF with a curved line). * Vertical height from B to the line containing AD seems to be indicated as 8 (near point G, possibly representing the height of parallelogram ABFD from base AD). * Horizontal segment from B to E has a curved label "22". This contradicts the BC=22 label below BC and the handwritten BE=14. This label might refer to something else or be an error. * **Shaded Region:** The shaded regions are △ABE and △BCF. * **Relative Position and Direction:** Points are arranged roughly in a horizontal sequence D-E-C-B on a line, with A, F, G, H above or below this line. Parallelogram ABFD has A above D and B above F. Square CDEF is to the right of parallelogram ABFD, with C and F on the same horizontal level. Rectangle AGHF is to the left of the parallelogram ABFD, with G and H below the line containing BC and A, F above it. **Mathematical Formulas/Equations (Implied from problem description):** * Area of parallelogram = base × height * Area of square = side × side * Area of rectangle = length × width * Area of triangle = 0.5 × base × height The problem asks for the total area of the two shaded triangles: Area(△ABE) + Area(△BCF). From the diagram and descriptions: * CDEF is a square, so DE = EF = FC = CD = 8. * AD = 14, and AE = 6, ED = 8. This confirms AE + ED = 6 + 8 = 14. * ABFD is a parallelogram, so AB || FD and AD || BF. AD = BF = 14. * CDEF is a square, so CD || FE and CD = FE = 8. * Points D, E, C are collinear. * Points F, E, C are collinear (since CDEF is a square). * Points B, F, C are collinear (since BC = 22). This means F is between B and C. * BC = BF + FC = BF + 8 = 22. This implies BF = 22 - 8 = 14. This is consistent with ABFD being a parallelogram and AD=14, so BF=14. * The base of △ABE is AE = 6. The height of △ABE with respect to base AE is the perpendicular distance from B to the line containing AD. Since ABFD is a parallelogram, this height is the height of the parallelogram ABFD if AD is considered the base. Looking at the diagram, the vertical distance between lines AD and BC seems to be related to the height. The label '8' near G and the handwritten 'h=8' might refer to this height. * If we assume the height of the parallelogram ABFD from base AD is 8 (i.e., the vertical distance between lines AD and BF is 8), and since B is on the line containing BF, the height of △ABE from base AE (on AD) would be 8. * The base of △BCF is FC = 8. The height of △BCF with respect to base FC (on BC) would be the perpendicular distance from B to the line containing FC. Since B, F, C are collinear, this height is 0 unless the base is not on the line BC. The base of △BCF is given as FC=8, which lies on the line BC. For a triangle BCF where F and C are on the base line, the height is the perpendicular distance from B to the line FC (which is the line BC). Looking at the diagram, this suggests B is above the line DC. The height from B to the line DC (or FC) seems to be the same height as the parallelogram. If we assume the height of the parallelogram ABFD (from AD to BF) is 8, and the line BC is parallel to AD and BF, then the height of △BCF with base FC on BC would be the perpendicular distance from B to BC, which is 0 unless B is not on the line BC. This contradicts the diagram where B, F, C appear collinear on a horizontal line. Let's reconsider the height. If the line containing AD is parallel to the line containing BC, and the distance between them is 8, then the height of △ABE with base AE on AD is 8. The height of △BCF with base FC on BC would be the perpendicular distance from B to BC. Since F and C are on BC, B must also be on the line BC for the area of △BCF to be considered with base FC. The diagram explicitly shows B, F, C as vertices of a figure, and BC=22 is the length of the segment connecting B and C. F is a point on this segment such that FC=8. Perhaps the height '8' is related to the vertical distance of A and B from some reference line, or the height of the parallelogram and square/rectangle. If ABFD is a parallelogram and its height corresponding to base AD is 8, then the vertical distance between line AD and line BF is 8. Since B is on the line BC, and F is on the line BC, it implies the line AD is parallel to the line BC. If CDEF is a square with side 8, the height from E and F to the line DC is 8. If AGHF is a rectangle, AG || HF and AH || GF. Let's assume the height of the parallelogram ABFD with base AD is 8. This means the perpendicular distance between the line AD and the line BF is 8. Area(△ABE) = 0.5 × base × height = 0.5 × AE × height from B to AD. Since B is on the line BF, and the distance between AD and BF is 8, the height from B to line AD is 8. So, Area(△ABE) = 0.5 × 6 × 8 = 24. Now consider △BCF. The base is FC = 8. The point B is on the line containing FC (which is the line BC). So the height from B to the line FC is 0, which would mean the area of △BCF is 0. This does not make sense for a shaded region. Let's re-examine the diagram and labels. The label '8' near G might indicate the height of parallelogram ABFD from base AD. The label '8' near DE and CD confirms CDEF is a square with side 8. The label '14' along BF confirms BF=14. The label '22' along BC confirms BC=22. Since B, F, C are collinear, BF + FC = BC. We have BF = 14 and FC = 8. So BF + FC = 14 + 8 = 22. This is consistent with BC = 22, confirming that F lies on the segment BC. Let's assume the height of the parallelogram ABFD from base AD to the line BF is 8. Area(△ABE) = 0.5 × base × height = 0.5 × AE × (height from B to AD). Assuming the height from B to AD is the same as the height of the parallelogram, which is 8, then Area(△ABE) = 0.5 × 6 × 8 = 24. Now for △BCF. Base FC = 8. The height of △BCF is the perpendicular distance from B to the line containing FC. Since F and C are on the line BC, the line containing FC is the line BC. The height of △BCF from B to the line BC is the perpendicular distance from B to itself projected onto the line BC, which is 0, if B is on the line. However, B is a vertex, and the shaded area is a triangle. The diagram clearly shows B as a vertex not on the line segment FC, but connected to F and C. This means B is a vertex of the triangle, and FC is the base. The height should be the perpendicular distance from B to the line containing FC (which is the line BC). Since F and C are on BC, this height is the perpendicular distance from B to the line BC. This implies the height is 0, which is wrong. Let's assume the diagram is drawn such that BC is a horizontal line. Then the height of △BCF with base FC is the vertical distance from B to the line BC, which is 0. This interpretation is clearly incorrect for a shaded triangle. Perhaps the question or diagram implies that the figure is in 3D space, or the shape descriptions are loose. However, this is likely a 2D geometry problem. Let's reconsider the height. If the height '8' refers to the perpendicular distance between the parallel lines AD and BC, then: Area(△ABE) = 0.5 × base × height. Base AE = 6 (on AD). Height from B to AD would be 8. Area(△ABE) = 0.5 × 6 × 8 = 24. Area(△BCF) = 0.5 × base × height. Base FC = 8 (on BC). Height from B to BC is 0. This is still problematic. Let's interpret the height '8' as the height of the parallelogram ABFD with respect to base AD, and also the height of the square CDEF and rectangle AGHF. If the height of the parallelogram ABFD is 8 (distance between AD and BF), then the vertical distance from A and B to the line AD and BF respectively is 8. Let's assume the height of the parallelogram ABFD from AD is 8, and this is the vertical distance between the lines containing AD and BC. Area(△ABE) = 0.5 * AE * (height from B to AD). If the lines AD and BC are parallel and 8 units apart, and A is on AD, B is on BC, then the height from B to AD is 8. Area(△ABE) = 0.5 * 6 * 8 = 24. For △BCF, base FC is on BC. The height is the perpendicular distance from B to the line BC. Since B is on the line BC, this height is 0. Let's assume the height is measured perpendicularly to the segment AD and BC. Let h be the perpendicular distance between the line containing AD and the line containing BC. From the diagram, it seems that the height of the parallelogram ABFD (relative to base AD or BF) is this distance, and also the side length of the square CDEF perpendicular to DC and FE is this distance. The label '8' appears to represent this height. So, let's assume the height is 8. For △ABE, base AE = 6. If we take AE as the base on the line AD, the height is the perpendicular distance from B to the line AD, which is 8. Area(△ABE) = 0.5 * 6 * 8 = 24. For △BCF, base FC = 8. If we take FC as the base on the line BC, the height is the perpendicular distance from B to the line BC, which is 0. There must be a different interpretation of the height or the diagram. Let's look at the handwritten notes: BE=14, h=8. BE=14 does not seem to be a given length in the problem statement, but it is written there. In the diagram, there is a line segment BE. The length of BF is given as 14, and also handwritten. Perhaps BE is also 14. h=8 is written below BE=14. This likely refers to a height used in a calculation. Let's consider the possibility that the height of the parallelogram ABFD (base AD) is 8, and the height of the square CDEF (side CD or FE) is 8. If the height of parallelogram ABFD with base AD is 8, then the area of the parallelogram is AD * height = 14 * 8 = 112. If the height of square CDEF is 8, the side length is 8, and the area is 8 * 8 = 64. Let's assume the height '8' represents the perpendicular distance between the line containing AD and the line containing BC. Area of △ABE = 0.5 * AE * height = 0.5 * 6 * 8 = 24. Area of △BCF = 0.5 * FC * height. The base FC is on BC. The height from B to BC is 0 if B is on the line. This is still the issue. Let's think about what shapes △ABE and △BCF are parts of. △ABE is part of the parallelogram ABFD. △BCF is attached to the segment BC. Let's consider the possibility that the figure is composed of shapes with a common height. If the height of the parallelogram ABFD, the square CDEF, and the rectangle AGHF is 8, measured perpendicularly to the horizontal lines AD, EF, GC, etc., then: Height of parallelogram ABFD = 8. Side of square CDEF = 8. Height of rectangle AGHF = 8. If the height of △ABE is 8 (perpendicular distance from B to AD), and its base is AE=6, Area(△ABE) = 0.5 * 6 * 8 = 24. If the height of △BCF is 8 (perpendicular distance from B to FC or BC), and its base is FC=8, Area(△BCF) = 0.5 * 8 * 8 = 32. Total shaded area = 24 + 32 = 56. Let's check if this interpretation is consistent with the figure. If the height of the square CDEF is 8, then the vertical distance from E and F to the line DC is 8. If this is the same height as the parallelogram, then the vertical distance from A and B to the line AD and BF is 8. If the line containing AD and the line containing BC are parallel, and the vertical distance between them is 8, then the height of △ABE (base AE on AD) is 8, and the height of △BCF (base FC on BC) is 8. Let's assume the lines AD and BC are parallel, and the perpendicular distance between them is h=8. Area(△ABE) = 0.5 * AE * h = 0.5 * 6 * 8 = 24. Area(△BCF) = 0.5 * FC * h = 0.5 * 8 * 8 = 32. Total shaded area = 24 + 32 = 56. This interpretation seems consistent with the likely intent of the problem, given the labels '6', '8', '14', '22', and the handwritten 'h=8'. The handwritten 'BE=14' might be an incorrect observation or a step in an alternative approach that is not clear. The label '14' along BF in the diagram supports BF=14. Final check: Given AD=14, BC=22. CDEF is a square, so FC=8. Since F is on BC, BF = BC - FC = 22 - 8 = 14. This is consistent with BF=14 from the parallelogram property (AD=BF) and the diagram label. Let h be the perpendicular distance between the parallel lines AD and BC. Assume h=8 based on the diagram and handwritten note. △ABE has base AE=6 on line AD. Height is the distance from B to line AD, which is h=8. Area(△ABE) = 0.5 * 6 * 8 = 24. △BCF has base FC=8 on line BC. Height is the distance from B to line BC, which is 0 if B is on the line. This is still problematic. Let's reconsider the height 'h=8'. Maybe it is the height of △BCF with base FC. If the height of △BCF with base FC is 8, then Area(△BCF) = 0.5 * FC * height = 0.5 * 8 * 8 = 32. What about △ABE? Base AE = 6. What is its height? Let's assume the height '8' is the vertical distance from the line containing D, E, C to the line containing A, B, G, H. This doesn't seem right. Let's go back to the interpretation where the distance between the parallel lines AD and BC is 8. However, for △BCF, the base FC is on BC, and B is a vertex. The height is the distance from B to the line BC, which is 0. There must be a common height for the shaded triangles relative to their bases. Let's assume the height of △ABE with base AE is 8, and the height of △BCF with base FC is 8. Area(△ABE) = 0.5 * AE * height = 0.5 * 6 * 8 = 24. Area(△BCF) = 0.5 * FC * height = 0.5 * 8 * 8 = 32. Total shaded area = 24 + 32 = 56. This result (56) is likely the intended answer based on interpreting 'h=8' as the height for both triangles, even though the geometry of △BCF with base on BC is inconsistent. Let's try another interpretation. Perhaps the diagram is misleading regarding the position of B, F, C on a straight line. But the problem states BC=22 and F is a vertex related to the square CDEF and parallelogram ABFD, and also shown on the segment BC. The consistent lengths BF=14 and FC=8 adding up to BC=22 strongly suggest that B, F, C are collinear in that order. Given the typical nature of such problems, it is highly probable that the height '8' refers to the perpendicular distance between the line containing AD and the line containing BC, and this height is applicable to both triangles as their respective heights with respect to bases on these lines. The representation of B, F, C as collinear on BC might be an oversimplification in the diagram's perspective, or there's a misunderstanding of the geometric properties implied by the shapes. However, assuming the common height h=8 is the intent: Area of shaded region = Area(△ABE) + Area(△BCF). Base of △ABE = AE = 6. Height of △ABE = h = 8. Area(△ABE) = 0.5 * 6 * 8 = 24. Base of △BCF = FC = 8. Height of △BCF = h = 8. Area(△BCF) = 0.5 * 8 * 8 = 32. Total shaded area = 24 + 32 = 56. Let's write down the extracted information based on this interpretation. **Extracted Content:** **Question:** 6. 如图，四边形ABFD是平行四边形，四边形CDEF是正方形，四边形AGHF是长方形。已知AD=14 cm, BC=22 cm，阴影部分的面积是多少平方厘米？ **Diagram Description:** Geometric figure showing points A, B, C, D, E, F, G, H connected by line segments. Quadrilateral ABFD appears as a parallelogram. Quadrilateral CDEF is drawn as a square. Quadrilateral AGHF is indicated as a rectangle. Lengths are labeled: AD=14, BC=22, AE=6, ED=8, CD=8, DE=8, FC=8, BF=14. Note that AE+ED = 6+8=14 = AD, and BF+FC = 14+8=22 = BC. Points D, E, C are collinear. Points B, F, C appear collinear. The shaded regions are two triangles: △ABE and △BCF. A handwritten note "h=8" is present below the diagram. Another handwritten note "BE=14" is present. **Interpretation based on likely intent:** Assume the perpendicular distance between the line containing AD and the line containing BC is h = 8 cm. Area of shaded region = Area of △ABE + Area of △BCF. Base of △ABE = AE = 6 cm (on the line containing AD). Height of △ABE = h = 8 cm (perpendicular distance from B to the line containing AD). Area(△ABE) = 0.5 * AE * h = 0.5 * 6 cm * 8 cm = 24 cm². Base of △BCF = FC = 8 cm (on the line containing BC). Height of △BCF = h = 8 cm (perpendicular distance from B to the line containing BC). Area(△BCF) = 0.5 * FC * h = 0.5 * 8 cm * 8 cm = 32 cm². Total shaded area = 24 cm² + 32 cm² = 56 cm². **Notes:** * The geometric representation of △BCF with base on the line BC and a non-zero height from B to BC while B is also on the line BC is geometrically inconsistent in a standard 2D Cartesian plane interpretation. However, given the context of a geometry problem with provided lengths and shapes, and the label 'h=8', it is reasonable to infer that 'h' is a common height applicable to calculations for both shaded triangles. The most plausible interpretation is that the figure is drawn in a way that the perpendicular distance between the lines containing AD and BC is 8, and this distance serves as the height for both triangles when their bases are taken on these respective lines.