Perimeter, Area & Volume — Quiz 5

Questions in this quiz (7)

1. A circular park has a walking path that follows an arc along its edge. The center of the park is labeled as $$C$$, and two points $$P$$ and $$Q$$ lie on the circle, forming an arc $$PQ$$. The sector $$CPQ$$ has an area of $$150\pi$$ square meters, and the central angle $$\angle PCQ$$ measures $$45^\circ$$. What is the length of arc $$PQ$$, in meters?

   • $$5\pi \sqrt{3}$$
   • $$20\sqrt{3}$$
   • $$40\pi\sqrt{3}$$
   • $$20\pi$$

2. A right triangular prism and a right triangular pyramid have the same base area but different heights. If the height of the pyramid is $$k$$ units, and the volume of the pyramid is $$\frac{1}{2}$$ the volume of the prism, what is the height of the prism in terms of $$k$$?

   • $$\frac{1}{2}k$$
   • $$\frac{1}{4}k$$
   • $$\frac{2}{3}k$$
   • $$\frac{5}{6}k$$

3. Right triangle XYZ has one angle that measures 60 degrees. If the longest side of the triangle is 6 units, what is the perimeter of the triangle?

   • $$6 + 3\sqrt{3}$$
   • $$9$$
   • $$6$$
   • $$9 + 3\sqrt{3}$$

4. Let the quadrilateral $$ABCD$$ be a right trapezoid such that: $$DC = 6$$, $$AB = x$$, and $$BC = 4$$Let triangle $$EFG$$ have a height $$EH$$ such that: $$EH = 3$$ and $$FG = 8x$$Determine the value of $$x$$ that makes the areas of trapezoid $$ABCD$$ and triangle $$EFG$$ equal.

   • $$\frac{6}{5}$$
   • $$\frac{5}{4}$$
   • $$6$$
   • $$\frac{4}{3}$$

5. If the lengths of all three sides of a triangle are decreased by 50%, what percent changes occur in its perimeter and area?

   • Perimeter: $$50\%$$ decrease Area: $$75\%$$ decrease
   • Perimeter: $$50\%$$ decrease Area: $$50\%$$ decrease
   • Perimeter: $$50\%$$ decrease Area: $$25\%$$ decrease
   • Perimeter: $$50\%$$ decrease Area: $$50\%$$ increase

6. Two equilateral triangles have side lengths of 1 unit and 4 units, respectively. How many of the smaller triangles are needed to completely fill the larger triangle without overlapping?

   • 12
   • 10
   • 16
   • 15

7. A square-shaped garden PQRS is connected to an equilateral triangular flower bed QRT. Point U is the midpoint of segment QR, and the length of segment UT is 8 cm. What is the total perimeter of the composite figure PQTRS?

   • 55.44
   • 45.54
   • 52.25
   • 48.27