QuanticaPrealgebraOpen in the courseFour-sided shapes get sorted into families the way biologists sort animals, by shared traits like parallel sides, equal sides, and right angles. Every member of a family inherits that family's properties, so once you know a shape is a parallelogram you get every parallelogram tool for free. We start with a frame of four rods bolted loosely at the corners.
Problem
A picture frame is built from two \(8\) inch rods and two \(5\) inch rods, hinged loosely at the corners. Push on one corner and the rectangle leans into the slanted shape above, then leans further with another push. The angles keep changing. What is the perimeter of the leaning shape, in inches?
Show a hint
- Leaning the frame does not change the rods, only the angles at the hinges.
- Add up the four rod lengths, \(8 + 5 + 8 + 5\).
Show the full solution
Add up the four rod lengths. \(8 + 5 + 8 + 5 = \boxed{26}\) inches. Leaning the frame changes the angles at the hinges, never the rods, so every position has this same perimeter.
Problem
In one leaning position, the hinge at the bottom left of the frame reads \(70\) degrees. The bottom rod and the top rod stay parallel, and so do the two side rods. Using what you know about parallel lines, how many degrees is the angle at the bottom right hinge?
Show a hint
- The two side rods are parallel, and the bottom rod crosses both of them, so the two bottom angles are co-interior angles.
- Co-interior angles between parallel lines add to \(180\) degrees.
Show the full solution
The two bottom hinge angles add to \(180\) degrees, so the right one is \(180 - 70 = \boxed{110}\) degrees. The bottom rod crosses both parallel side rods, which is what makes those two angles co-interior.
Problem
A parallelogram-shaped stencil has a base of \(10\) cm, slanted sides of \(7\) cm, and a height of \(6\) cm measured straight up from the base. Snip the overhanging triangle off one end and slide it across to plug the gap at the other end. The pieces form a rectangle. What is the stencil's area in square centimeters?
Show a hint
- The rectangle you build has the same base and the same height as the stencil.
- A \(10\) by \(6\) rectangle has area \(10 \times 6\).
Show the full solution
The rectangle is \(10\) by \(6\), so the area is \(10 \times 6 = \boxed{60}\) square centimeters. The snipped triangle fills the gap exactly, so nothing is gained or lost, and the \(7\) cm slant never enters the calculation.
Problem
A solar panel shaped like a parallelogram has sides of \(12\) m and \(5\) m. The perpendicular distance between its two \(12\) m edges is \(4\) m. An installer multiplies \(12 \times 5 = 60\) and quotes the area as \(60\) square meters. What is the actual area, in square meters?
Show a hint
- Area needs the base times the height, and the height is the perpendicular distance, not the slanted side.
- The base is \(12\) and the height is \(4\).
Show the full solution
Area is base times height, \(12 \times 4 = \boxed{48}\) square meters. The \(5\) m side is slanted, so it is longer than the true height of \(4\) m, and using it overquotes by \(12\) square meters.
Problem
A club logo is drawn by marking a horizontal segment of length \(12\) and a vertical segment of length \(9\) that cross at each other's midpoints, then connecting the four endpoints. The result is the rhombus above, a quadrilateral with four equal sides. It sits inside a dashed \(12\) by \(9\) rectangle, and in each quarter of that rectangle the logo covers exactly half. What is the area of the logo?
Show a hint
- The dashed rectangle has area \(12 \times 9 = 108\).
- Each quarter-rectangle is cut along its diagonal, so the logo takes exactly half of the whole rectangle.
Show the full solution
The logo covers half the dashed rectangle, and \(12 \times 9 = 108\), so the area is \(\boxed{54}\). Each quarter of the rectangle is cut along its diagonal, which is why the logo takes exactly half.
Problem
A rhombus-shaped garden plot has an area of \(40\) square feet. One diagonal measures \(10\) feet. Using the pattern you just discovered, how many feet long is the other diagonal?
Show a hint
- A rhombus fills half its bounding rectangle, so area \(=\) (diagonal \(\times\) diagonal) \(\div\, 2\).
- Solve \(10 \times d \div 2 = 40\).
Show the full solution
The diagonals multiply to twice the area, \(10 \times d = 80\), so \(d = \boxed{8}\) feet.
Problem
A highway warning sign is a rhombus with diagonals of \(12\) inches and \(16\) inches. The diagonals cross at right angles at their midpoints, cutting the sign into four identical right triangles. Use one of those triangles and the Pythagorean Theorem to find the side length of the sign, in inches.
Show a hint
- Each right triangle has legs equal to half of each diagonal, \(6\) and \(8\).
- Find the hypotenuse of a \(6\)-\(8\) right triangle.
Show the full solution
Half-diagonals of \(6\) and \(8\) form the legs, and the sign's side is the hypotenuse. Since \(6^2 + 8^2 = 36 + 64 = 100\), the side is \(\sqrt{100} = \boxed{10}\) inches.
Problem
A square napkin is folded once along its diagonal, and the crease measures \(10\) inches. A square is a rhombus whose diagonals are equal, so both diagonals of the napkin are \(10\). What is the napkin's area in square inches?
Show a hint
- Use the rhombus area tool with \(d_1 = d_2 = 10\).
- Half of \(10 \times 10\).
Show the full solution
With equal diagonals of \(10\), the rhombus tool gives \(\tfrac{1}{2} \times 10 \times 10 = \boxed{50}\) square inches. You met this napkin in the last lesson, where the side came out to \(\tfrac{10}{\sqrt{2}}\), and squaring that side gives the same \(50\).
Problem
A kitchen backsplash uses trapezoid tiles with parallel edges of \(5\) inches and \(11\) inches, set \(4\) inches apart. Tiles snap together in pairs, one flipped upside down as shown, forming a parallelogram. What is the area of that two-tile parallelogram, in square inches?
Show a hint
- The flipped tile lays its \(5\) inch edge alongside the other tile's \(11\) inch edge, so the parallelogram's base is \(5 + 11\).
- Base \(16\), height \(4\), area \(= 16 \times 4\).
Show the full solution
Snapped together, the pair forms a parallelogram with base \(5 + 11 = 16\) and the same \(4\) inch height, so the area is \(16 \times 4 = \boxed{64}\) square inches.
Problem
One single tile is half of that two-tile parallelogram. Check the shortcut on a bigger tile. A trapezoid window has parallel edges of \(9\) feet and \(13\) feet, set \(6\) feet apart. What is its area in square feet?
Show a hint
- Two copies would form a parallelogram of base \(9 + 13 = 22\) and height \(6\), and the window is half of it.
- Compute \(\tfrac{1}{2} \times 22 \times 6\).
Show the full solution
Doubling the window gives a parallelogram of area \(22 \times 6 = 132\), and one window is half of that, \(\tfrac{1}{2} \times 132 = \boxed{66}\) square feet. In one line, area \(= \tfrac{1}{2}(9+13)(6)\).
Practice these ideas
Practice
Here are four claims. (a) Every square is a rhombus. (b) Every rhombus is a square. (c) Every rectangle is a parallelogram. (d) A quadrilateral can have exactly three right angles. How many of the four claims are true?
Show the solution
(a) is true, a square's four equal sides make it a rhombus. (b) is false, a leaning rhombus has no right angles. (c) is true, right angles do not remove parallel sides. (d) is false, three right angles use up \(270\) degrees and the fourth must be \(90\), giving four right angles, never exactly three. That makes \(\boxed{2}\) true claims.
Practice
One angle of a parallelogram measures \(48\) degrees. How many degrees is the largest angle of the parallelogram?
Show the solution
A neighbor of the \(48\) degree angle measures \(180 - 48 = 132\) degrees, and opposite angles repeat the pair. The largest is \(\boxed{132}\).
Practice
A rhombus has a perimeter of \(52\) cm, and one of its diagonals measures \(24\) cm. Use the four right triangles inside the rhombus to find the length of the other diagonal, in centimeters.
Show the solution
Each side is \(52 \div 4 = 13\), and the half-diagonals are the legs of a right triangle with hypotenuse \(13\). One leg is \(24 \div 2 = 12\), so the other satisfies \(a^2 + 12^2 = 13^2\), giving \(a^2 = 169 - 144 = 25\) and \(a = 5\). The full diagonal is \(2 \times 5 = \boxed{10}\) cm.
Practice
A stained-glass pane is a rhombus with diagonals of \(14\) inches and \(6\) inches. What is its area in square inches?
Show the solution
Half the product of the diagonals is \(\tfrac{1}{2} \times 14 \times 6 = \boxed{42}\) square inches.
Practice
A square coaster has a diagonal of \(6\) inches. What is its area in square inches?
Show the solution
Treat the square as a rhombus with equal diagonals. Its area is \(\tfrac{1}{2} \times 6 \times 6 = \boxed{18}\) square inches, no side length needed.
Practice
A parallelogram-shaped field has an area of \(91\) square meters and a base of \(13\) meters. What is its height, in meters?
Show the solution
Height \(= 91 \div 13 = \boxed{7}\) meters.
Practice
The cross-section of a drainage channel is a trapezoid with parallel edges of \(4\) feet and \(10\) feet, set \(5\) feet apart. What is the area of the cross-section in square feet?
Show the solution
The average base is \(\tfrac{4+10}{2} = 7\), and \(7 \times 5 = \boxed{35}\) square feet.
Practice
A trapezoid banner has an area of \(60\) square inches, a height of \(6\) inches, and one base of \(7\) inches. How many inches long is the other base?
Show the solution
The sum of the bases is \(\tfrac{2 \times 60}{6} = 20\), so the other base is \(20 - 7 = \boxed{13}\) inches.
Practice
A pixel-art gem on a screen has corners at \((0,4)\), \((6,0)\), \((12,4)\), and \((6,8)\). Its diagonals are the horizontal segment from \((0,4)\) to \((12,4)\) and the vertical segment from \((6,0)\) to \((6,8)\). What is the gem's area in square pixels?
Show the solution
The diagonals measure \(12\) and \(8\), cross at \((6,4)\), and are perpendicular, so the area is \(\tfrac{1}{2} \times 12 \times 8 = \boxed{48}\) square pixels.
Practice
A tile has four \(6\) cm sides and at least one right angle. Explain to yourself which family the tile must belong to, then find its area in square centimeters.
Show the solution
Four equal sides make the tile a rhombus, hence a parallelogram. One \(90\) degree angle forces each neighbor to \(180 - 90 = 90\), so all four angles are right and the tile is a square. Its area is \(6^2 = \boxed{36}\) square centimeters.