Four squares are joined together to form one large square. If the perimeter of one of the original squares was 8x units, what is the perimeter of the new, larger square?
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Solution
The perimeter of each small square is 8x units; therefore, the length of a side of each small square is \(\frac{8x}{4}\) = 2x units. Since the new, large square is comprised of two sides from each of the four squares (the remaining four sides are now within the large square), the perimeter of the new, large square is equal to: 4(2x + 2x) = 4(4x) = 16x units.
If the perimeter of a parallelogram ABCD is equal to the perimeter of rhombus EFGH and the perimeter of square IJKL, then
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Solution
Since the rhombus and the square have the same perimeter and both figures have four congruent sides, every side of the rhombus and every side of the square is equal to exactly one-fourth of the perimeter. Therefore, every side of the rhombus must be congruent to every side of the square.
The length of a rectangle is four times the length of a square. If the rectangle and the square share a side, and the perimeter of the square is 2 m, what is the perimeter of the rectangle?
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Solution
If the square and the rectangle share a side, then the width of the rectangle is equal to the length of the square. If the length of the square is x, then x + x + x + x = 2, 4x = 2, x = 1⁄2. Since the width of the rectangle is equal to the length of the square and the length of the rectangle is 4 times the length of the square, the width of the rectangle is 1⁄2 and the length is 4(1⁄2) = 2. Therefore, the perimeter of the rectangle is equal to: 1⁄2 + 2 + 1⁄2 + 2 = 5 units.
The length of one side of a rhombus is x2 – 6. If the perimeter of the rhombus is 168 units, what is the value of x?
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Solution
Every side of a rhombus is equal in length. One side of this rhombus is equal to \(\frac{168}{4}=42\) units. Therefore, x2 – 6 = 42, x2 = 48, x = \(\sqrt{48}\) = \(\sqrt{16}\sqrt{3}\) = 4√3.
The length of a rectangle is four less than twice its width. If x is the width of the rectangle, what is the perimeter of the rectangle?
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Solution
If x is the width of the rectangle, then 2x – 4 is the length of the rectangle. Since opposite sides of a rectangle are congruent, the perimeter of the rectangle is equal to 2x – 4 + x + 2x – 4 + x = 6x – 8.
Monica draws a quadrilateral whose diagonals form four right triangles inside the quadrilateral. This quadrilateral
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Solution
The diagonals of a rhombus are perpendicular.A rhombus has four congruent sides, but its diagonals are not necessarily congruent, and only the opposite angles of a rhombus are necessarily congruent. If a quadrilateral has perpendicular diagonals, it must have four congruent sides.
The diagonals of rectangle ABCD intersect at E. Which of the following is NOT always true?
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Solution
Angles DCE and ECB are complementary angles; these angles combine to form right angle DCB of the rectangle. These angles are only equal if ABCD is a square. However, ABCD is a rectangle, and not all rectangles are squares. Therefore, it is not always true that angle DCE = angle ECB.
DeDe has four identical line segments. Using them, she can form
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Solution
In order to form a square or a rhombus, you must have four identical line segments. Since nothing about the angles is stated in the question, these four line segments could be connected at right angles to form a square. Since a square is a type of rectangle and a type of rhombus, both of which are types of parallelograms, any of these four types of quadrilaterals could be formed.
Angle E of rhombus EFGH measures 3x + 5. If the measure of angle H is 4x, what is the measure of angle F?
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Solution
Angles E and H are consecutive angles in rhombus EFGH. Therefore, their measures are supplementary: 3x + 5 + 4x = 180, 7x + 5 = 180, 7x = 175, x = 25. The measure of angle H is 4(25) = 100. Since opposite angles of a rhombus are congruent and angles H and F are opposite angles, angle F is also 100°.
Andrew constructs a polygon with four sides and no right angles. His polygon
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Solution
Andrew’s polygon has four sides, so it must be a quadrilateral. It does not contain a right angle, so it cannot be a rectangle or a square, since these quadrilaterals each have four right angles. However, this polygon could be a parallelogram or a rhombus, since these are four-sided polygons that do not necessarily contain a right angle.