Geometry — Quiz 8

Questions in this quiz (8)

1. A ramp is designed such that its incline ratio is $$\frac{1}{4}$$ (meaning it rises 1 foot vertically for every 4 feet horizontally). If the ramp forms a right triangle where the hypotenuse is the ramp length and it needs to be $$3\sqrt{10}$$ feet long, what are its approximate dimensions $$x$$ (vertical height) and $$y$$ (horizontal length)?

   • 0.25 , 9.2
   • 0.25 , 90
   • 2.3 , 90
   • 2.3 , 9.2

2. A triangle $$ABC$$ has sides $$AB=700$$ units, $$BC=600$$ units, and angle $$\angle B=10^\circ$$. What is the length of side $$AC$$?

   • $$600\tan(10^\circ)$$
   • $$\sqrt{700^2+600^2-2(700)(600)\cos(10^\circ)}$$
   • $$700^2+600^2-2(700)(600)\cos(10^\circ)$$
   • $$\sqrt{700^2+600^2}$$

3. Jason has a $$20\times14$$ inches rectangular piece of cardboard. He cuts $$3 \text{ inch} x 3 \text{ inch}$$ squares from each of the four corners and then folds up the sides by $$90^\circ$$ to form a rectangular box. How many $$1\text{-inch}$$ cubes can fit inside the resulting box?

   • $$280$$
   • $$112$$
   • $$224$$
   • $$336$$

4. Square WXYZ has a perimeter of 64 units. If points P, Q, R, and S are the midpoints of the sides WX, XY, YZ, and ZW respectively, what is the perimeter of square PQRS?

   • $$32\sqrt{2}$$ units
   • $$3\sqrt{2}$$ units
   • $$2\sqrt{3}$$ units
   • $$16\sqrt{2}$$ units

5. A unit circle is inscribed in $$\triangle ABC$$. In the triangle, $$\angle A = 60^\circ$$. Connect the circle center $$O$$ with vertex $$A$$. Find the length of segment $$AO$$.

   • $$2$$
   • $$1$$
   • $$\frac{1}{2}$$
   • $$4$$

6. In the figure below, triangle $$ABC$$ is a right triangle where $$\angle A = 30^\circ$$ and the hypotenuse $$AB = 10$$. Triangle $$DEF$$ is a right triangle where $$\angle D = 45^\circ$$ and the hypotenuse $$DE = 10$$. What is the total length of line segment $$AE$$?

   • $$5(\sqrt{3} + \sqrt{2})$$
   • $$3 \sqrt{3}$$
   • $$5 \sqrt{2}$$
   • $$2 \sqrt{5}$$

7. Two chords and a diameter form a triangle inside a circle. The radius is $$5$$ cm and one chord is $$2$$ cm longer than the other one. Find the perimeter and the area of the triangle.

   • Perimeter = $$12$$ cm Area = $$8$$ cm²
   • Perimeter = $$12$$ cm Area = $$10$$ cm²
   • Perimeter = $$10$$ cm Area = $$12$$ cm²
   • Perimeter = $$24$$ cm Area = $$24$$ cm²

8. Two vertices of a triangle PQR are $$(2,5)$$ and $$(8,-1)$$. If the coordinates of the centroid of triangle PQR are $$(6,3)$$, find the coordinates of the third vertex R.

   • $$(4,9)$$
   • $$(10,3)$$
   • $$(8,5)$$
   • $$(3,4)$$