Quadrilaterals — Hard Practice Quiz

A Geometry cheat sheet for Quadrilaterals — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. A square has a perimeter $P_S$. A rectangle has a perimeter $P_R$ such that $P_R = P_S$. If the length of the rectangle is three times its width, what is the ratio of the area of the square to the area of the rectangle, $A_S / A_R$?

   • $4/3$
   • $3/4$
   • $1$
   • $2/3$

   Answer: $4/3$

2. A rhombus has diagonals $d_1$ and $d_2$. A square has a diagonal equal to $d_1$. If the area of the rhombus is equal to the area of the square, what is the relationship between $d_1$ and $d_2$?

   • $d_2 = d_1$
   • $d_2 = 2d_1$
   • $d_2 = d_1 / 2$
   • $d_2 = d_1 \sqrt{2}$

   Answer: $d_2 = d_1$

3. A parallelogram has sides $a$ and $b$ and an angle $\theta$ between them. A square has a perimeter equal to the perimeter of this parallelogram. If the area of the parallelogram is $A_P = ab\sin\theta$, what is the area of the square, $A_S$, in terms of $a$, $b$, and $\theta$?

   • $\frac{(a+b)^2}{4}$
   • $ab\sin\theta$
   • $\frac{a^2+b^2}{4}$
   • $\frac{a^2+b^2+2ab\sin\theta}{4}$

   Answer: $\frac{(a+b)^2}{4}$

4. A trapezoid has bases $b_1$ and $b_2$ and height $h$. A rectangle is constructed such that its length is the average of the trapezoid's bases, and its width is the trapezoid's height. How does the area of this rectangle, $A_R$, compare to the area of the trapezoid, $A_T$?

   • $A_R = A_T$
   • $A_R = 2A_T$
   • $A_R = A_T / 2$
   • $A_R = A_T \sqrt{2}$

   Answer: $A_R = A_T$

5. A kite has diagonals $d_1$ and $d_2$, and the angle between them is $\theta$. If this kite is also a rhombus, what is the relationship between its area formula $A = \frac{1}{2}d_1 d_2 \sin\theta$ and the general rhombus area formula $A = \frac{1}{2}d_1 d_2$?

   • The kite formula simplifies to the rhombus formula because $\sin\theta = 1$ for a rhombus
   • The formulas are always different
   • The kite formula is only for non-rhombic kites
   • The rhombus formula is a special case of the kite formula when $d_1 = d_2$

   Answer: The kite formula simplifies to the rhombus formula because $\sin\theta = 1$ for a rhombus

6. A cyclic quadrilateral has sides $a, b, c, d$. If this quadrilateral is a rectangle, what does Brahmagupta's formula $A = \sqrt{(s-a)(s-b)(s-c)(s-d)}$ simplify to?

   • $lw$
   • $l^2+w^2$
   • $\frac{1}{2}lw$
   • $\sqrt{l^2+w^2}$

   Answer: $lw$

7. If the diagonal of a square is increased by a factor of $k$, by what factor does its area increase?

   • $k^2$
   • $k$
   • $\sqrt{k}$
   • $2k$

   Answer: $k^2$

8. A rectangle has a fixed perimeter $P$. What is the maximum possible area it can have, and what shape is the rectangle when its area is maximized?

   • $A = \frac{P^2}{16}$, a square
   • $A = \frac{P^2}{4}$, a square
   • $A = \frac{P^2}{16}$, a rectangle with $l=2w$
   • $A = \frac{P^2}{8}$, a square

   Answer: $A = \frac{P^2}{16}$, a square

9. A trapezoid has bases $b_1$ and $b_2$ and height $h$. If $b_1 = b_2 = b$, what does the trapezoid area formula $A = \frac{1}{2}(b_1 + b_2)h$ simplify to, and what shape does the trapezoid become? How does this simplified formula relate to the parallelogram area formula $A = bh$?

   • It simplifies to $A = bh$, becoming a parallelogram, which is the parallelogram area formula
   • It simplifies to $A = bh$, becoming a rectangle, which is the parallelogram area formula
   • It simplifies to $A = \frac{1}{2}bh$, becoming a triangle
   • It remains $A = \frac{1}{2}(b_1 + b_2)h$ as it's still a trapezoid

   Answer: It simplifies to $A = bh$, becoming a parallelogram, which is the parallelogram area formula

10. A rhombus has a perimeter $P$. If one of its diagonals is $d_1$, what is its area in terms of $P$ and $d_1$?

    • $\frac{d_1}{4}\sqrt{P^2 - 4d_1^2}$
    • $\frac{d_1}{2}\sqrt{P^2 - d_1^2}$
    • $\frac{P}{4}\sqrt{d_1^2 - P^2}$
    • $\frac{P d_1}{4}$

    Answer: $\frac{d_1}{4}\sqrt{P^2 - 4d_1^2}$