ACT 1 Geo: Understand relationships between

Welcome to your ACT 1 Geo: Understand relationships between

$Q_{1}:$ Let $ABCDEF \sim KLMNOP$. If the perimeter of the first hexagon is $66$ ft and the perimeter of the second is $72$ ft, what is the ratio of their areas?

$121:144$

$11:12$

$144:120$

$66:70$

$Q_{2}:$ The surface areas of two similar right cylinders are $80\pi$ $in^{3}$ and $120\pi $ $in^{3}$, respectively. What is the ratio of their volumes?

$8: 12$

$2 : 3 $

$\sqrt{2} : \sqrt{3} $

$2\sqrt{2} : 3\sqrt{3} $

$Q_{3}:$ A rectangular prism has a length of $15$cm, a width of $9 $cm, and a height of $6 $cm. If each dimension is divided by $3$, what is the ratio of the volume of the original prism to the second prism?

$27:1$

$3:1$

$9:1$

$1:9$

$Q_{4}:$ If two similar octagons have a scale factor of $3:5$, then the ratio of their areas is

$3:5$

$\sqrt{3} : \sqrt{5}$

$9:25$

$27:125$

$Q_{5}:$ A special garden design requires that the garden have three distinct square sections whose areas follow the ratio $2:3:5$. If such a garden is designed to have a total area of $1550$ square feet, then what would be the area of the smallest section in square feet?

$155$

$500$

$250$

$300$

$Q_{6}:$ Each side of square $A$ has a length of $3$ meters, while each side of square $B$ has a length of $9$ meters. What is the ratio of the area of square $A$ to the area of square $B$?

$1:9$

$1:6$

$1:3$

$1:1$

$Q_{7}:$ A rectangle has sides of length $x$ and $x + 1$, where $x$ is a positive number. If the area of the rectangle is $12$, then which of the following is equivalent to the ratio of $x$ to $x + 1$?

$2:3$

$3:4$

$1:3$

$1:4$

$Q_{8}:$ If the sides of two rectangles are in the ratio $4 : 8$, what is the ratio of their areas?

$1:2$

$1:3$

$1:4$

$1:5$

$Q_{9}:$ In the figure shown below, the four squares have the same center. What is the ratio of the perimeter of the outermost square to the perimeter of the innermost square?

$5:4$

$4:1$

$4:5$

$1:4$

$Q_{10}:$ In the $xy$‑plane below, a point (not shown) with coordinates $(a, b)$ lies on the graph of the linear function $h$. If $a$ and $b$ are positive integers, what is the ratio of $a$ to $b$?

$2:3$

$3:2$

$5:6$

$6:5$

$Q_{11}:$ Isosceles trapezoid $JKLM$ is shown below. If the dimensions of trapezoid $JKLM$ are multiplied by a scale factor of f to create trapezoid $J′K′L′M′$, which statement is true?

Trapezoid $J′K′L′M′$ contains two base angles measuring $30$ each.

The longer base of trapezoid $J′K′L′M′$ is $56f$ units.

The bases of trapezoid $J′K′L′M′$ have lengths of $22$ units and $39$ units.

Trapezoid $J′K′L′M′$ contains two base angles measuring $(120f )$ each.

$Q_{12}:$ If a sphere and a right circular cone have the same radius and equal volumes, what is the ratio of the height of the cylinder to its radius?

$4$

$\frac{1}{4}$

$\frac{\pi }{3}$

$\frac{3\pi }{4}$

$Q_{13}:$ What is the ratio of the circumference of a circle to its radius?

$1$

$\frac{\pi }{2}$

$\sqrt{\pi }$

$2\pi $

$Q_{14}:$ In rhombus $PQRS$, the ratio of $m\angle P$ to $m \angle Q$ is $1$ to $5$ and $PQ = 6$. What is the area of the rhombus?

$6$

$12$

$18$

$24$

$Q_{15}:$ If the interior angles of a pentagon are in the ratio of $2 : 3 : 3 : 5 : 5$, what is the measure of the smallest angle?

$20$

$40$

$60$

$80$

$Q_{16}:$ In the diagram below, the circle is inscribed in square $ABCD$, and square $PQRS$ is inscribed in the circle. What is the ratio of the area of the large square to the area of the small square?

$2:1$

$2:3$

$2:4$

$8:2$

$Q_{17}:$ In the figure below, the ratio of the measure of arc $AB$ to the measure of arc $BC$ to the measure of arc $CA$ is $4$ to $3$ to $5$. What is $m\angle A$?

$30$

$45$

$60$

$75$

$Q_{18}:$ In the figure below, circle $O$ is tangent to sides $\overline{AD}$ and $\overline{BC}$ of rectangle $ABCD$. If the area of the shaded region is $3$ times the area of the circle, what is the ratio of the length of side $\overline{AD}$ to the circumference of the circle?

$\frac{3}{\pi }$

$1$

$\frac{\pi }{3}$

$3$

$Q_{19}:$ The base of pyramid $1$ is a $3-4-5$ triangle, and the base of pyramid $2$ is a square whose sides are $3$. If the volumes of the pyramid are equal, what is the ratio of the height of pyramid $1$ to the height of pyramid $2$?

$\frac{4}{9}$

$\frac{2}{3}$

$\frac{4}{3}$

$\frac{3}{2}$

$Q_{20}:$ If the volume of a sphere is equal to the volume of a cube, what is the ratio of the edge of the cube to the radius of the sphere?

$1.16$

$1.33$

$1.53$

$1.61$

$Q_{21}:$ In Figure below shows two right circular cylinders, $C$ and $C′$. If $r = kr′$ and $h = kh′$, then what is the ratio of Volume of $C$ to Volume of $C'$?

$\frac{1}{\pi }$

$\pi$

$k \pi$

$k^{3} $

$Q_{22}:$ In the circle above, chord $RS$ is parallel to diameter $PQ$. If the length of $RS$ is $\frac{3}{ 4}$ of the length of $PQ$ and the distance between the chord and the diameter is $2 \sqrt{7}$, what is the radius of the circle?

$2$

$3\sqrt{7}$

$8$

$4\sqrt{7}$

$Q_{23}:$ The length and width of a large picture are respectively $18$ inches and $12$ inches. If each dimension is reduced by $x$ inches to make the ratio of length to width $5$ to $3$, what is the value of $x $?

$6$

$5$

$4$

$3$

$Q_{24}:$ If triangle $ABC$ is similar to triangle $DEF$ and the segments are given as marked, then $EH$ is equal to ?