Geometry Drills: Problems, Answers, and Explanations

Practicing Geometry through Drills

Geometry

Basic geometry may be defined as dealing with the characteristics and connections of lines and points, surfaces and solids, as well as other spatial relationships. Geometry has a number of subsections. The subsections below (not all-inclusive) contain sample problems, answers and explanations.

Plane Geometry

Plane geometry includes lines, planes and polygons. Conceptually, lines and planes do not have ends or edges and extend indefinitely, even though we draw points, lines and planes that look like they have size limitations.

Problem

True or False? - One way to form two lines is to intersect two planes.

Answer and Explanation

False. Planes are flat surfaces that extend indefinitely in all directions, so if they intersect, only one line is formed at the place of intersection.

Solid Geometry

Solid geometry deals with 3-dimensional figures. These include prisms, spheres, pyramids, cones and cylinders. Because all of their sides are flat, these shapes may be grouped together in the category of polyhedrons, which also includes other shapes that have more sides (e.g., tetrahedron and octahedron).

Problem

You have a cone and a cylinder. The radius of each is five inches, and their height is also equal at 11 inches. What is the ratio of their volumes?

Answer and Explanation

The ratio of the cylinder's volume to the cone's volume is 3:1. Use the formulas for the volumes of a cone and a cylinder.

The volume of a cylinder is (pi)(radius squared)(height). Plug in the numbers so the problem looks like this: V = (3.14)(5^2)(11) = 863.5 cubic inches.

The volume of a cone is (1/3)(pi)(radius squared)(height). Use the information from the problem to solve: V = (1/3)(3.14)(5^2)(11) = 287.8 cubic inches.

As a result, the ratio is 863.5 cubic inches to 287.8 cubic inches, which can be reduced to 3:1.

Loci

A locus is a group of points (or lines) that are decided by or satisfy defined conditions. You solve a locus problem by drawing a diagram.

Problem

A private detective is searching for the spot where he believes his client's stolen jewels may have been buried. He has information about a rock that is 20 feet from a particular pine tree and ten feet from a certain oak tree. He has deduced that the jewels are the same distance from the rock and the oak tree, and that they are also eight feet from the oak tree. How many possible places will the detective have to look to find the jewels?

Answer and Explanation

He will have to look in two places. Draw a diagram with the rock, pine tree and oak tree. The rock and the pine tree must be 20 feet from one another. The oak tree may be placed anywhere that is ten feet from the rock in your drawing, although the detective has an exact location for it.

Around the oak tree, draw a circle with a radius of eight feet. Draw a straight line between the rock and the pine tree. Bisect it with a perpendicular line. The line will cross the circumference of the circle in two places. If the detective's deductions are accurate, the jewels will be in one of the two places.

Proofs

Giving geometric proofs is basically an exercise in logic. Proofs use pre-defined theorems to prove solutions to problems. The problem below is only a small representation of the many geometric proofs.

Problem

True or False? - A triangle can have sides measuring nine, 11 and 18.

Answer and Explanation

True. The pertinent theorem states that if you add any two sides of a triangle, the sum must be larger than the third side. In this case, 9 + 11 = 20, which is greater than 18. Similarly, 11 + 18 = 29, which is greater than nine. Finally, 9 + 18 = 27, which is greater than 11. Since each of the sums of two sides is greater than the third side, nine, 11 and 18 can be the lengths of the sides of a triangle.

Transformations

Transformational geometry is used to understand congruence and similarity by moving a shape, while keeping the same size and proportions. Movements made include flips (or reflections), slides (or glides or translations) and rotations (or turns). A flip is a mirror image; a slide is what it sounds like - sliding the shape to a different place on the plane. A rotation is the figure turned to a different angle. Problems and their solutions are given in terms of coordinates.

Problem

True or False? - Using the translation (2, -3), the point (-5, 4) will become (-3, 1).

Answer and Explanation

True. The translation tells you to add two to the x-value (-5) and subtract three from the y-value (4).