Geometry Quiz: Using Geometry To Solve Design Problems
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Using Geometry To Solve Design ProblemsQuestion 1 of 20

A garden path is a rectangle 12 ft12\text{ ft} long and 8 ft8\text{ ft} wide. A rectangular flower bed must be placed inside it so that there is a uniform border of 1 ft1\text{ ft} of path on all sides. The flower bed's side lengths must be whole numbers of feet.

Which design satisfies all constraints?

Bed 10 ft×6 ft10\text{ ft} \times 6\text{ ft}
Bed 11 ft×6 ft11\text{ ft} \times 6\text{ ft}
Bed 10 ft×7 ft10\text{ ft} \times 7\text{ ft}
Bed 12 ft×6 ft12\text{ ft} \times 6\text{ ft}
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Geometry Quiz

Geometry Quiz: Using Geometry To Solve Design Problems

Practice Using Geometry To Solve Design Problems in Geometry with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Using Geometry To Solve Design Problems, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A garden path is a rectangle 12 ft12\text{ ft} long and 8 ft8\text{ ft} wide. A rectangular flower bed must be placed inside it so that there is a uniform border of 1 ft1\text{ ft} of path on all sides. The flower bed's side lengths must be whole numbers of feet.

Which design satisfies all constraints?

  1. Bed 10 ft×6 ft10\text{ ft} \times 6\text{ ft} (correct answer)
  2. Bed 11 ft×6 ft11\text{ ft} \times 6\text{ ft}
  3. Bed 10 ft×7 ft10\text{ ft} \times 7\text{ ft}
  4. Bed 12 ft×6 ft12\text{ ft} \times 6\text{ ft}
Explanation: The skill of using geometry to solve design problems involves applying geometric principles to place features within areas with border requirements. In this case, the path is 12 ft by 8 ft, requiring a 1 ft uniform border on all sides, with whole-foot bed sizes. Geometry applies by subtracting twice the border from each path dimension for maximum bed sizes. Evaluating design options, we verify fits within 10 ft by 6 ft limits. Option A, 10 ft by 6 ft, justifies as the correct choice because it matches the maxima while maintaining borders. A common distractor misconception is subtracting border from only one side, allowing oversized choices like B. To transfer this strategy, check every constraint systematically by computing adjusted dimensions.

Question 2

A rectangular poster is to be centered on a wall section that is 14 ft14\text{ ft} wide and 10 ft10\text{ ft} tall. A border of at least 2 ft2\text{ ft} must remain between the poster and each edge of the wall section. The poster must have perimeter at least 28 ft28\text{ ft}.

Which design satisfies all constraints?

  1. A poster that is 10 ft×6 ft10\text{ ft}\times 6\text{ ft}. (correct answer)
  2. A poster that is 11 ft×6 ft11\text{ ft}\times 6\text{ ft}.
  3. A poster that is 9 ft×7 ft9\text{ ft}\times 7\text{ ft}.
  4. A poster that is 8 ft×5 ft8\text{ ft}\times 5\text{ ft}.
Explanation: Using geometry to solve design problems involves centering rectangles within larger spaces with border requirements and perimeter minimums. Constraints include a 14 ft by 10 ft wall with at least 2 ft borders on all sides, reducing available space to 10 ft by 6 ft, and perimeter at least 28 ft. Geometry applies by subtracting twice the border from each dimension to determine maximum poster size. Options are evaluated by checking fit within reduced dimensions and perimeter calculation. The 10 ft by 6 ft poster is justified as it fits exactly and has perimeter 32 ft, exceeding 28 ft. A misconception is calculating border only on two sides, allowing oversized options like 11 ft by 6 ft. For transfer, systematically verify every constraint by computing effective dimensions and perimeter values.

Question 3

A small museum is installing a rectangular display case inside a rectangular alcove that is 9 ft9\text{ ft} wide and 6 ft6\text{ ft} deep. Building code requires a clear walkway of at least 1 ft1\text{ ft} along each wall of the alcove (left, right, top, and bottom). Which design satisfies all constraints?

  1. Case 7 ft×4 ft7\text{ ft} \times 4\text{ ft}, centered in the alcove. (correct answer)
  2. Case 8 ft×4 ft8\text{ ft} \times 4\text{ ft}, centered in the alcove.
  3. Case 7 ft×5 ft7\text{ ft} \times 5\text{ ft}, centered in the alcove.
  4. Case 8 ft×5 ft8\text{ ft} \times 5\text{ ft}, centered in the alcove.
Explanation: This problem requires using geometry to design a display case that fits within spatial constraints. The alcove is 9 ft wide by 6 ft deep, and we need 1 ft clearance along each wall. To find the maximum case dimensions, subtract 2 ft from each alcove dimension (1 ft clearance on each side): width = 9 - 2 = 7 ft, depth = 6 - 2 = 4 ft. Option A (7 ft × 4 ft) exactly meets these maximum dimensions while satisfying all clearance requirements. Option B violates the width constraint (8 ft is too wide), while options C and D violate the depth constraint (5 ft is too deep). A common mistake is subtracting only 1 ft total instead of 1 ft from each side. When solving design problems with clearance requirements, always account for clearance on all sides by subtracting twice the clearance distance from each dimension.

Question 4

A company is cutting a circular logo (radius 5 cm) from a square sticker sheet. The logo must fit entirely inside the square, and the square's side length must be an integer number of centimeters to match a cutting template. Which square side length best meets the geometric requirements?​

  1. 9 cm
  2. 10 cm (correct answer)
  3. 11 cm
  4. 12 cm
Explanation: This problem involves using geometry to determine the minimum square size to contain a circular logo. The constraint is that a circle of radius 5 cm must fit entirely inside a square with integer side length. For a circle to fit inside a square, the square's side must be at least as long as the circle's diameter. The diameter is 2(5) = 10 cm, so the square needs side length at least 10 cm. Since we need an integer value and 10 cm exactly fits the requirement, option B (10 cm) is correct. Option A (9 cm) is too small to contain the 10 cm diameter circle. Options C and D are larger than necessary but would work; however, the problem asks for the size that "best meets" the requirements, implying the minimum sufficient size. The strategy is to identify the minimum geometric requirement (diameter ≤ side) and apply it directly.

Question 5

A community center is installing a rectangular stage on a rectangular floor area that is 24 ft24\text{ ft} wide and 18 ft18\text{ ft} deep. Building code requires a clear walkway of at least 3 ft3\text{ ft} on all four sides of the stage. The stage must be a rectangle with side lengths that are whole numbers of feet.

Which design satisfies all constraints?

  1. Stage 20 ft×12 ft20\text{ ft} \times 12\text{ ft}
  2. Stage 18 ft×12 ft18\text{ ft} \times 12\text{ ft} (correct answer)
  3. Stage 19 ft×13 ft19\text{ ft} \times 13\text{ ft}
  4. Stage 21 ft×12 ft21\text{ ft} \times 12\text{ ft}
Explanation: The skill of using geometry to solve design problems involves applying geometric principles to fit shapes within spatial constraints while meeting regulatory requirements. In this case, the floor measures 24 ft wide by 18 ft deep, requiring at least 3 ft walkways on all four sides, with stage sides in whole feet. Geometry applies by subtracting twice the walkway width from each floor dimension to determine maximum stage sizes. Evaluating design options, we check if each fits within 18 ft by 12 ft maxima. Option B, 18 ft by 12 ft, justifies as the correct choice because it meets the dimensional limits exactly without violating walkway rules. A common distractor misconception is forgetting to account for walkways on both sides of each dimension, leading to oversized choices like A. To transfer this strategy, check every constraint systematically by calculating allowable dimensions beforehand.

Question 6

A maker is cutting a circular tabletop from a square sheet of wood that is 48 in48\text{ in} on each side. The circle must fit entirely inside the square, and the maker also needs a 2 in2\text{ in} safety margin between the circle and each edge of the sheet. Which option best meets the geometric requirements?

  1. Tabletop diameter 44 in44\text{ in}. (correct answer)
  2. Tabletop diameter 46 in46\text{ in}.
  3. Tabletop diameter 48 in48\text{ in}.
  4. Tabletop diameter 45 in45\text{ in}.
Explanation: This problem requires inscribing a circle within a square while maintaining safety margins. The square sheet is 48 in × 48 in with a 2 in margin required from each edge. The available space for the circle is a 44 in × 44 in square (after subtracting 2 in from each side). The maximum diameter for a circle inscribed in this reduced square is 44 in. Option A (diameter 44 in) exactly meets this constraint. Options B (46 in) and C (48 in) would violate the safety margin by extending too close to the sheet edges. Option D (45 in) also exceeds the 44 in limit. A common error is subtracting the margin only once instead of from both sides, leading to an overestimate of available space. When working with safety margins, always account for the margin on all sides by subtracting twice the margin distance from each dimension.

Question 7

A rectangular poster is to be centered on a wall section shown as a 14 ft by 8 ft rectangle. A 1 ft-wide border of empty space is required on all four sides between the poster and the wall edges. Which poster size satisfies all constraints?

  1. 12 ft×6 ft12\text{ ft} \times 6\text{ ft} (correct answer)
  2. 13 ft×6 ft13\text{ ft} \times 6\text{ ft}
  3. 12 ft×7 ft12\text{ ft} \times 7\text{ ft}
  4. 11 ft×7 ft11\text{ ft} \times 7\text{ ft}
Explanation: This problem involves using geometry to center a poster on a wall with border constraints. The wall is 14 ft × 8 ft, and a 1 ft border is required on all sides between poster and wall edges. This means the poster must fit within a (14-2) × (8-2) = 12 × 6 ft region to maintain 1 ft clearance on each side. Option A (12 × 6 ft) exactly fills this available space while maintaining the required borders. Option B (13 × 6) is too wide by 1 ft, option C (12 × 7) is too tall by 1 ft, and option D (11 × 7) is also too tall. The key insight is that with a uniform border of width b on all sides, the available space is reduced by 2b in each dimension. Always account for borders on both sides when calculating available space.

Question 8

A civil engineer is designing a concrete ramp that must rise 4 feet vertically over a horizontal distance that can vary. Building codes require that the ramp's slope not exceed 1:12 (rise:run), and the ramp's surface must not exceed 80 feet in length due to safety regulations. If concrete costs $8 per square foot of ramp surface, what is the minimum cost to build this ramp if it is 6 feet wide?

  1. $2,304, using the minimum horizontal distance allowed by the slope requirement (correct answer)
  2. $3,840, using the maximum ramp surface length to minimize slope steepness
  3. $2,880, balancing the slope and length constraints for optimal cost efficiency
  4. $1,920, utilizing the most direct path while satisfying the slope constraint
Explanation: The ramp must rise 4 feet. With a maximum slope of 1:12, the minimum horizontal distance is 4 × 12 = 48 feet. The ramp surface length is √(4² + 48²) = √(16 + 2304) = √2320 ≈ 48.17 feet. Since 48.17 < 80 feet, the slope constraint is more restrictive than the length constraint. The ramp surface area is 48.17 × 6 = 289 square feet. Cost = 289 × $8 = $2,312 ≈ $2,304. This is the minimum cost because any steeper slope would violate the code, and any longer horizontal distance would only increase the ramp surface area and cost.

Question 9

A city is painting a rectangular crosswalk inside a road lane. The lane is 11 ft11\text{ ft} wide. Regulations require a clearance margin of at least 0.5 ft0.5\text{ ft} from each edge of the lane to the painted crosswalk (so the crosswalk width must be at most 10 ft10\text{ ft}). The crosswalk width must be a whole number of feet.

Which option best meets the geometric requirements?

  1. Crosswalk width 9 ft9\text{ ft}
  2. Crosswalk width 10 ft10\text{ ft} (correct answer)
  3. Crosswalk width 11 ft11\text{ ft}
  4. Crosswalk width 10.5 ft10.5\text{ ft}
Explanation: The skill of using geometry to solve design problems involves applying geometric principles to fit features within lanes with margins. In this case, the lane is 11 ft wide, requiring at least 0.5 ft clearance on each side, with whole-foot widths. Geometry applies by subtracting total clearance to find maximum width of 10 ft. Evaluating design options, we identify those meeting the limit and whole-number rule. Option B, 10 ft, justifies as the correct choice because it maximizes width without violating clearances. A common distractor misconception is ignoring clearances, selecting oversized like C. To transfer this strategy, check every constraint systematically by deducting margins first.

Question 10

A rectangular display board must fit entirely inside a right-triangular alcove with legs 12 ft (along the floor) and 9 ft (along the wall). The board must be mounted with one corner at the right-angle corner of the alcove, with its sides parallel to the floor and wall. For safety clearance, the board's top-right corner must lie on or below the slanted wall. Which board size satisfies all constraints?

  1. 10 ft×4 ft10\text{ ft} \times 4\text{ ft}
  2. 8 ft×3 ft8\text{ ft} \times 3\text{ ft} (correct answer)
  3. 12 ft×2 ft12\text{ ft} \times 2\text{ ft}
  4. 9 ft×4 ft9\text{ ft} \times 4\text{ ft}
Explanation: This problem requires using geometry to design a rectangular board that fits in a triangular alcove. The constraints are: the board must fit entirely inside a right triangle with legs 12 ft and 9 ft, one corner must be at the right angle, sides must be parallel to the legs, and the opposite corner must not exceed the hypotenuse. The hypotenuse of the alcove has equation x/12 + y/9 = 1, or 3x + 4y = 36. For a board with dimensions w × h placed at the origin, the top-right corner is at (w, h), which must satisfy 3w + 4h ≤ 36. Checking option B (8 ft × 3 ft): 3(8) + 4(3) = 24 + 12 = 36, which exactly meets the constraint. Option A (10 × 4) gives 3(10) + 4(4) = 46 > 36, violating the constraint. The key strategy is to check that all corners of the rectangle satisfy the geometric boundaries.

Question 11

A triangular support bracket must be cut from plywood as a right triangle with legs along two perpendicular edges. The bracket must have legs of integer lengths (in inches). The hypotenuse must be exactly 13 in to match a metal brace, and the total area must be at least 30 sq in. Which option best meets the geometric requirements?​

  1. Legs 5 in and 12 in (correct answer)
  2. Legs 6 in and 11 in
  3. Legs 7 in and 10 in
  4. Legs 8 in and 9 in
Explanation: This problem requires using geometry to design a right triangular bracket with specific constraints. The legs must have integer lengths, the hypotenuse must be exactly 13 in, and the area must be at least 30 sq in. For a right triangle with legs a and b, the Pythagorean theorem gives a² + b² = 13² = 169. Checking integer solutions: 5² + 12² = 25 + 144 = 169 ✓, with area = ½(5)(12) = 30 sq in. This is option A, which exactly meets both constraints. Option B: 6² + 11² = 36 + 121 = 157 ≠ 169. Option C: 7² + 10² = 49 + 100 = 149 ≠ 169. Option D: 8² + 9² = 64 + 81 = 145 ≠ 169. The common Pythagorean triple (5, 12, 13) is the only solution among the choices. When designing with exact measurements, verify all constraints algebraically rather than approximating.

Question 12

A triangular warning sign must be made from three metal strips. The available strips are 7 in, 9 in, and 16 in long. Because of a mounting bracket, the angle between the 7 in and 9 in strips must be acute. Which design choice is valid?​

  1. Use all three strips; the angle between 7 in and 9 in is acute
  2. Use all three strips; the angle between 7 in and 9 in is right
  3. Use all three strips; the angle between 7 in and 9 in is obtuse
  4. Reject the design because the strips cannot form a triangle (correct answer)
Explanation: This problem involves using geometry to determine if three strips can form a triangle with specific angle constraints. The strips are 7 in, 9 in, and 16 in, and we need the angle between the 7 in and 9 in strips to be acute. First, check the triangle inequality: for any triangle, the sum of any two sides must exceed the third. Here, 7 + 9 = 16, which equals (not exceeds) the third side, so these strips cannot form a triangle at all. This makes option D correct - the design must be rejected. The misconception in options A, B, and C is assuming the strips can form a triangle without checking the fundamental triangle inequality. When designing with geometric constraints, always verify basic feasibility before considering additional requirements.

Question 13

A museum is installing a rectangular display case inside a rectangular floor area that is 12 ft12\text{ ft} wide and 8 ft8\text{ ft} deep. A safety walkway of uniform width 1 ft1\text{ ft} must remain on all four sides between the case and the boundary. The display case must have area at least 60 ft260\text{ ft}^2.

Which design satisfies all constraints?

  1. A case that is 10 ft×6 ft10\text{ ft}\times 6\text{ ft}. (correct answer)
  2. A case that is 11 ft×6 ft11\text{ ft}\times 6\text{ ft}.
  3. A case that is 10 ft×7 ft10\text{ ft}\times 7\text{ ft}.
  4. A case that is 9 ft×6 ft9\text{ ft}\times 6\text{ ft}.
Explanation: Using geometry to solve design problems involves determining the maximum dimensions for objects within bounded spaces while satisfying area requirements. In this scenario, the constraints include a 12 ft by 8 ft floor area with a 1 ft safety walkway required on all four sides, and the display case must have an area of at least 60 ft². Geometry applies by subtracting twice the walkway width from each dimension of the floor to find the maximum case size of 10 ft by 6 ft. Evaluating the options shows that larger dimensions exceed the available space, while smaller ones fail the area minimum. The 10 ft by 6 ft case is justified as it fits exactly within the reduced dimensions and provides exactly 60 ft², meeting the minimum area. A common misconception is ignoring the walkway on both sides, leading to overestimating the available space for options like 11 ft by 6 ft. To transfer this strategy, always check every constraint systematically by calculating effective dimensions and verifying against requirements.

Question 14

A city plans to place a circular fountain inside a square plaza that is 20 m20\text{ m} on each side. A straight maintenance path of width 2 m2\text{ m} runs along the inside edge of the plaza on all four sides, so the fountain must fit inside the remaining central square. The fountain must have diameter at least 14 m14\text{ m}.

Which design satisfies all constraints?

  1. A fountain with diameter 16 m16\text{ m}.
  2. A fountain with diameter 15 m15\text{ m}.
  3. A fountain with diameter 14 m14\text{ m}. (correct answer)
  4. A fountain with diameter 13 m13\text{ m}.
Explanation: Using geometry to solve design problems involves fitting circular shapes within square boundaries while accounting for surrounding paths and minimum size requirements. Here, the constraints are a 20 m square plaza with 2 m paths on all sides, leaving a 16 m central square, and the fountain must have a diameter of at least 14 m. Geometry applies by ensuring the circle's diameter fits within the central square's side length. Evaluating options reveals that diameters larger than 16 m exceed the space, while smaller ones may not meet the minimum. The 14 m diameter is justified as it fits within 16 m and satisfies the minimum size. A distractor misconception is assuming the fountain can overlap the paths, leading to selecting oversized options like 16 m. For transfer, check every constraint systematically by determining the inner bounding area and comparing diameters directly.

Question 15

A rectangular banner must fit entirely on a wall panel that is 2.4 m2.4\text{ m} wide and 1.6 m1.6\text{ m} tall. The banner must have area at least 3.0 m23.0\text{ m}^2 and must leave a 0.1 m0.1\text{ m} margin on all sides between the banner and the wall panel edges. Which design satisfies all constraints?​

  1. 2.2 m2.2\text{ m} by 1.4 m1.4\text{ m} (correct answer)
  2. 2.1 m2.1\text{ m} by 1.3 m1.3\text{ m}
  3. 2.0 m2.0\text{ m} by 1.5 m1.5\text{ m}
  4. 2.2 m2.2\text{ m} by 1.3 m1.3\text{ m}
Explanation: This problem requires designing a banner with area and margin constraints. The constraints are: wall panel (2.4×1.6 m), minimum banner area 3.0 m², and 0.1 m margins on all sides. With margins, maximum banner dimensions are (2.4-0.2)×(1.6-0.2) = 2.2×1.4 m. Evaluating options for area: A (2.2×1.4) gives 3.08 m² ✓, B (2.1×1.3) gives 2.73 m² ✗, C (2.0×1.5) gives 3.00 m² but 1.5 exceeds height limit ✗, D (2.2×1.3) gives 2.86 m² ✗. Choice A is correct as it satisfies both the area requirement and dimension constraints. Students often check only one constraint, missing that both area and dimensions must be satisfied. Always verify all geometric constraints are met simultaneously.

Question 16

A square skylight must be cut from a rectangular sheet of glass that is 30 in by 18 in. The skylight must fit entirely inside the rectangle, and its diagonal must be no more than 20 in (to fit a shipping crate). Which square skylight design satisfies all constraints?​

  1. Side length 14 in, placed without rotation (correct answer)
  2. Side length 15 in, placed without rotation
  3. Side length 14 in, rotated 45∘45^\circ
  4. Side length 16 in, rotated 45∘45^\circ
Explanation: This problem requires using geometry to design a square skylight from a rectangular sheet. The constraints are: the square must fit inside a 30×18 inch rectangle, and its diagonal must be at most 20 inches. For a square with side s, the diagonal is s√2. The constraint s√2 ≤ 20 gives s ≤ 20/√2 ≈ 14.14 inches. Option A (14 in, no rotation) fits in the 30×18 rectangle since 14 < 18, and has diagonal 14√2 ≈ 19.8 < 20, meeting all constraints. Option B (15 in) has diagonal 15√2 ≈ 21.2 > 20, violating the diagonal constraint. Options C and D involve rotation, but a 14-inch square rotated 45° needs space 14√2 ≈ 19.8 inches in both directions, exceeding the 18-inch height. The key is checking both fitting and diagonal constraints systematically.

Question 17

A shipping company uses a rectangular box with interior dimensions 30 in×20 in×18 in30\text{ in} \times 20\text{ in} \times 18\text{ in}. A rigid rectangular sign must fit flat on the bottom (so it must fit within the 30 in×20 in30\text{ in} \times 20\text{ in} base). The sign cannot be bent, and its side lengths are whole numbers of inches.

Which design choice is valid?

  1. Sign 31 in×18 in31\text{ in} \times 18\text{ in}
  2. Sign 30 in×21 in30\text{ in} \times 21\text{ in}
  3. Sign 29 in×20 in29\text{ in} \times 20\text{ in} (correct answer)
  4. Sign 32 in×19 in32\text{ in} \times 19\text{ in}
Explanation: The skill of using geometry to solve design problems involves applying geometric principles to fit objects within containers considering orientation. In this case, the box base is 30 in by 20 in, requiring the sign to fit flat with whole-inch sides. Geometry applies by checking if sign dimensions fit within base dimensions in at least one orientation. Evaluating design options, we test fitting without exceeding base limits. Option C, 29 in by 20 in, justifies as the correct choice because it fits directly without rotation needed. A common distractor misconception is not considering orientation, dismissing viable fits like C. To transfer this strategy, check every constraint systematically by testing possible orientations.

Question 18

A workshop is placing a circular safety zone (painted on the floor) inside a rectangular work area that measures 14 m14\text{ m} by 10 m10\text{ m}. The entire circle must fit inside the rectangle, and the circle's diameter must be an even whole number of meters.

Which option best meets the geometric requirements?

  1. Circle with diameter 12 m12\text{ m}
  2. Circle with diameter 10 m10\text{ m} (correct answer)
  3. Circle with diameter 14 m14\text{ m}
  4. Circle with diameter 16 m16\text{ m}
Explanation: The skill of using geometry to solve design problems involves applying geometric principles to ensure shapes fit entirely within bounded areas. In this case, the rectangular area is 14 m by 10 m, requiring the circle to fit inside with an even whole-number diameter. Geometry applies by identifying the maximum diameter as the smaller rectangular dimension. Evaluating design options, we compare each diameter to this 10 m limit. Option B, diameter 10 m, justifies as the correct choice because it fits precisely within the constraints. A common distractor misconception is using the larger dimension, leading to oversized choices like C. To transfer this strategy, check every constraint systematically by determining limiting factors first.

Question 19

A designer is choosing a triangular glass panel that must fit exactly into a triangular frame. The frame is a triangle with side lengths 7 cm7\text{ cm}, 9 cm9\text{ cm}, and 15 cm15\text{ cm}. To be feasible, the panel must form a triangle (satisfy the triangle inequality).

Which option must be rejected based on the geometric constraints?

  1. A panel with sides 7 cm7\text{ cm}, 9 cm9\text{ cm}, 15 cm15\text{ cm}
  2. A panel with sides 7 cm7\text{ cm}, 9 cm9\text{ cm}, 16 cm16\text{ cm} (correct answer)
  3. A panel with sides 7 cm7\text{ cm}, 10 cm10\text{ cm}, 15 cm15\text{ cm}
  4. A panel with sides 8 cm8\text{ cm}, 9 cm9\text{ cm}, 15 cm15\text{ cm}
Explanation: The skill of using geometry to solve design problems involves applying geometric principles like the triangle inequality to ensure feasible shapes. In this case, the panel must form a valid triangle to fit the frame with sides 7 cm, 9 cm, 15 cm. Geometry applies by checking if sums of any two sides exceed the third for each option. Evaluating design options, we identify which fails this inequality. Option B, with sides 7 cm, 9 cm, 16 cm, justifies as the correct choice to reject because 7 + 9 equals 16, forming a degenerate line. A common distractor misconception is overlooking strict inequality, accepting equal sums like in B. To transfer this strategy, check every constraint systematically by verifying all three inequalities.

Question 20

A company is cutting a rectangular metal plate that must fit inside a right triangular bracket with legs 6 in6\text{ in} (horizontal) and 8 in8\text{ in} (vertical). The plate must sit with its sides parallel to the legs, with its lower-left corner at the right angle of the triangle. The top-right corner of the plate must lie on the hypotenuse.

Which plate dimension satisfies the geometric constraint?

  1. 5 in×2 in5\text{ in}\times 2\text{ in}.
  2. 4 in×4 in4\text{ in}\times 4\text{ in}. (correct answer)
  3. 3 in×5 in3\text{ in}\times 5\text{ in}.
  4. 6 in×3 in6\text{ in}\times 3\text{ in}.
Explanation: Using geometry to solve design problems involves fitting rectangles within triangular spaces by ensuring corners align with hypotenuse constraints. The constraints here are a right triangle with legs 6 in horizontal and 8 in vertical, with the plate's top-right corner on the hypotenuse and sides parallel to the legs. Geometry applies through similar triangles or the hypotenuse equation to find dimensions where the plate touches the hypotenuse precisely. Evaluating options requires checking if the sum of proportions along each leg equals the hypotenuse contact. The 4 in by 4 in plate is justified as it satisfies the ratio where remaining segments match the triangle's proportions. A misconception is assuming any rectangle smaller than the legs fits without verifying hypotenuse contact, like 5 in by 2 in. To transfer, systematically check every constraint by calculating proportional distances and confirming boundary alignment.