Geometry Practice Problems with Step-by-Step Answers

This page gives mixed geometry practice problems with answers. Try each problem first, then open the step-by-step solution, and check that the answer is reasonable: units match, angles sum correctly, and lengths scale consistently. Common errors are unit mismatch, misreading the diagram, and applying the wrong transformation rule. These are worked examples, not official Regents or EOC tests.

Before You Practice: How to Use This Page

Six problems across five skill areas: reading a diagram, triangles, a proof, area and perimeter, and transformations. Work through every problem below before scrolling to the answer key; the answers sit together in one solution block so you cannot peek early. Each problem points to the concept it tests, and the wider concept map lives in the geometry concepts guide, which this practice page does not repeat.

Mixed Practice Problems and Answers

Read the Diagram and Terms

Use notation and the distance formula. Problem 1. Points A(1, 2) and B(4, 6) mark the endpoints of a segment. Find the length AB. Jot down your number before you open the answer key below. Do not read the solution block yet; give yourself three quiet minutes.

Triangles and Shapes

Problem 2. The three angles of a triangle are in the ratio 1 : 2 : 3. Find each angle. Problem 3. Triangle ABC has AB = AC. Explain, using symmetry, why the angles opposite those equal sides must be equal. Write two or three sentences.

Area and Perimeter

Problem 4. A rectangle is 12 long and 5 wide. Find its perimeter and its area. Perimeter is the distance around the shape, and area is the flat space it covers. Label which number is perimeter and which is area before you check.

Transformations

Problem 5. Reflect the point P(3, −2) across the x-axis and write the coordinates of its image. Problem 6. Dilate the triangle A(1, 1), B(3, 1), C(1, 4) by scale factor k = 2 about the origin. Write the image vertices and the new area.

Step-by-Step Solutions and Answer Key

Problem 1. Answer: AB = 5. Steps: the straight-line distance is √((4 − 1)² + (6 − 2)²) = √(3² + 4²) = √25 = 5. Check: 3, 4, 5 is a standard right-triangle triple, so the result fits. Common trap: adding 3 + 4 = 7, the taxicab distance. If you got 7, you skipped squaring the differences and taking the root.

Problem 2. Answer: 30°, 60°, 90°. Steps: call the angles x, 2x, and 3x. Because angles sum to 180°, x + 2x + 3x = 180°, so 6x = 180° and x = 30°; the angles are 30°, 60°, and 90°. Check: 30 + 60 + 90 = 180, and the parts keep the 1 : 2 : 3 order. Common trap: dividing 180 by 1 + 2 + 3 = 6 and forgetting to multiply the other parts.

Problem 3. Answer: ∠B = ∠C. Steps: fold the triangle along the angle bisector of ∠A. Since AB = AC, the two halves match edge for edge, so point B lands on point C. The angles at B and C land on each other, so they are equal. Check: a triangle with two equal sides folds onto itself along the bisector. Common trap: saying the angles are equal because they look equal in the sketch; a drawing is not a reason. Triangle classification is developed in the geometric shapes guide.

Problem 4. Answer: perimeter 34, area 60. Steps: perimeter = 2 × (12 + 5) = 34; area = 12 × 5 = 60. Check: 2 × 12 + 2 × 5 = 24 + 10 = 34, and the area is in square units. Common trap: writing 12 + 12 + 5 = 29, which counts only one width. Perimeter adds all four sides.

Problem 5. Answer: P′(3, 2). Steps: reflection across the x-axis keeps the x-coordinate and flips the sign of the y-coordinate, so (3, −2) lands at (3, 2). Check: the image sits the same distance below the axis as the original sits above, on the same vertical line. Common trap: flipping the wrong coordinate, which is what reflection across the y-axis does, giving (−3, −2).

Problem 6. Answer: A′(2, 2), B′(6, 2), C′(2, 8), new area 12 square units. Steps: about the origin, (x, y) becomes (kx, ky), so the vertices double. The legs along the axes are 2 and 3, giving area (1/2) × 2 × 3 = 3. The dilated legs are 4 and 6, giving area (1/2) × 4 × 6 = 12. Check: area scales by k², and 3 × 2² = 3 × 4 = 12. Common trap: scaling the area by k instead of k², which would give 6.

Notation such as AB and AB̄ is covered in the geometry terms reference. The reflection geometry guide, rotation geometry guide, and dilation geometry guide cover the transformation rules used here.

Checking Your Work

After each problem, run three checks. Units first: a length answer carries length units, an area answer carries square units. Sense next: angles in a triangle sum to 180°, and a shrunken figure stays smaller than the original. Last, reverse the operation: work backward from your answer and recover the given numbers. The question-plus-solution pairing follows the worked-solution style of BYJU'S geometry questions; for continued practice on one topic, the unit-based exercises on Khan Academy are the next stop.

Common Errors and How to Diagnose Them

• Unit mismatch: the question asks for area and you return a length. Diagnosis: circle the asked-for quantity before solving.
• Misread diagram: treating a marked tick as equal length when it carries another meaning. Diagnosis: re-read the marks before naming any side.
• Wrong transformation rule: mixing reflection across the x-axis with reflection across the y-axis. Diagnosis: sketch the new point and compare sides.
• Wrong scale-factor direction: enlarging when the factor is below 1. Diagnosis: check whether k is above or below 1 before drawing.

The geometry diagnostic questions page goes deeper on error causes.

Where to Practice Next

Pick the topic where your check failed: terms and notation for Problem 1, triangles and proof for Problems 2 and 3, area for Problem 4, transformations for Problems 5 and 6. Redo that problem with fresh numbers, then mix two topics at once.

The Gauth Geometry course includes practice exercises organized by unit, which matches this practice set. To drill at your own level, open Gauth Math Courses and enter geometry practice problems.

Sources and Review

• Khan Academy — High School Geometry, reviewed 2026-10-28. Unit-based course practice across congruence, similarity, and transformations.
• BYJU'S — Geometry Questions with Solutions, reviewed 2026-10-28. Question-plus-step-solution pairing style.