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Quizzes > Mathematics

Pythagorean Theorem Right Triangle Test: Quick Practice

Quick geometry quiz with right triangle practice. Instant results.

Editorial: Review CompletedCreated By: Jeff MossUpdated Aug 27, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art style illustration of right triangle, geometric shapes, Pythagoras formula and quiz prompt on sky blue background

This quiz helps you practice the Pythagorean theorem and find missing sides in right triangles. Get clear questions with instant feedback and step-by-step checks. For more focused review, try the right triangle test and the pythagorean theorem test, or drill special cases with the special right triangles quiz.

What is the length of the hypotenuse of a right triangle with legs of length 3 and 4?
4.5
5
7
6
According to the Pythagorean theorem, the hypotenuse c satisfies c² = 3² + 4² = 9 + 16 = 25, so c = 5. This fundamental result is the simplest Pythagorean triple example. Knowing this basic (3,4,5) triangle helps in many geometry problems and proofs.
A right triangle has a hypotenuse of length 10 and one leg of length 6. What is the length of the other leg?
7
4
6.5
8
Using c² = a² + b², we have b = ?(10² ? 6²) = ?(100 ? 36) = ?64 = 8. The Pythagorean theorem always applies to right triangles for finding a missing side. Once you know two sides, you can solve for the third using this relationship.
Does the triangle with side lengths 5, 12, and 13 form a right triangle?
Only if it is equilateral
No
Yes
Only if it is isosceles
For a right triangle, a² + b² must equal c² for the largest side. Here 5² + 12² = 25 + 144 = 169 and 13² = 169, so the triangle is right. Recognizing common triples like (5,12,13) is useful in many applications.
Is the triangle with sides 7, 24, and 25 a right triangle?
No
It depends on the angles
Only if one angle is 60°
Yes
Checking 7² + 24² = 49 + 576 = 625 and 25² = 625 shows the triangle is right by the Pythagorean theorem. Large integer triples like (7,24,25) follow the same rule as smaller ones. Identifying these triples saves time in problem solving.
Which of these formulas expresses the Pythagorean theorem for a right triangle with legs a and b and hypotenuse c?
a + b = c
a² = b² ? c²
c² = a² + b²
c = a² + b²
The Pythagorean theorem states that for any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, c² = a² + b². This equation is foundational in Euclidean geometry. It only holds when the angle between sides a and b is exactly 90°.
What is the distance between the points (0, 0) and (12, 5) in the coordinate plane?
12 + 5
13
17
?144 + 25
The distance formula is ?[(12?0)² + (5?0)²] = ?(144 + 25) = ?169 = 13. This is a direct application of the Pythagorean theorem in analytic geometry. It calculates the straight-line distance between two points.
In an isosceles right triangle, the hypotenuse measures 14. What is the length of each leg?
7?2
14/2
14?2
7
In a 45°-45°-90° triangle, legs are equal and the hypotenuse is leg × ?2. So each leg = 14/?2 = 7?2. Recognizing special right triangles simplifies many geometry problems. Always rationalize if exact form is required.
What is the length of the diagonal of a square with side length 8?
8
8?2
16
4?2
A square's diagonal forms a right triangle with sides 8 and 8, so diagonal = ?(8² + 8²) = ?128 = 8?2. This special case of the Pythagorean theorem is often used in design and engineering. Remember that diagonals in squares are equal in length.
In a right triangle, the side opposite the 30° angle is half the hypotenuse. If the hypotenuse is 10, what is the length of the side opposite 30°?
2?3
5
10?3
5?3
A 30°-60°-90° triangle has sides in the ratio 1 : ?3 : 2. The side opposite 30° is 1 × (hypotenuse/2), so 10/2 = 5. Recognizing these ratios speeds up calculations.
Calculate the distance between the points (3, 4) and (7, 1).
5
?34
6
4
Use the distance formula: ?[(7?3)² + (1?4)²] = ?(16 + 9) = ?25 = 5. This applies the Pythagorean theorem in the coordinate plane. It finds the straight-line distance between any two points.
A ladder leans against a wall. The foot is 10 ft from the wall and the ladder reaches 26 ft up. How long is the ladder?
26 ft
20 ft
16 ft
24 ft
Treat the ladder as the hypotenuse c of a right triangle with legs 10 and the unknown height. Here the known vertical reach is 26, so actually the ladder is the hypotenuse and the height is 26, so the base must be ?(c² ? 26²) = 10 ? c = ?(10² + 26²) = ?(100 + 676) = ?776 ?27.86. But if the ladder reaches 26, then the ladder length is the hypotenuse: ?(10² + 26²) = ?776 ?27.86. (Correcting: the problem intended ladder = hypotenuse; use legs 10 and height x: x = ?(26² ? 10²) = 24.)
In a right triangle, an altitude to the hypotenuse divides it into segments of lengths 9 and 16. What is the length of the altitude?
15
12
?225
?25
The altitude h to the hypotenuse in a right triangle satisfies h = ?(segment? × segment?) = ?(9 × 16) = 12. This geometric mean relation is a standard result in right triangle geometry. It arises from similar triangles formed by the altitude.
Classify the triangle with side lengths 6, 7, and 10 as acute, right, or obtuse.
Isosceles
Acute
Right
Obtuse
Compare the square of the longest side to the sum of squares of the other sides: 10² = 100, 6² + 7² = 36 + 49 = 85. Since 100 > 85, the triangle is obtuse. This is a direct application of the converse Pythagorean theorem.
Two legs of a right triangle differ by 2, and the hypotenuse is 10. What are the lengths of the legs?
7 and 9
4 and 6
6 and 8
5 and 7
Let the legs be a and a+2. Then a² + (a+2)² = 100 gives 2a² + 4a + 4 = 100 ? a² + 2a ? 48 = 0 ? a = 6 (positive). Thus legs are 6 and 8. This algebraic approach extends the Pythagorean theorem.
Find the inradius of a right triangle whose legs measure 15 and 20.
15/4
5
10/3
7.5
The inradius r of any triangle is r = (a + b ? c) / 2, where c is the hypotenuse. Here c = ?(15² + 20²) = 25, so r = (15 + 20 ? 25) / 2 = 5. This formula is often derived from area relationships.
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Study Outcomes

  1. Understand Pythagorean Theorem fundamentals -

    Gain a clear grasp of how the sides of a right triangle relate through the formula c² = a² + b².

  2. Apply the theorem to calculate missing sides -

    Use the Pythagorean Theorem to solve for unknown legs or the hypotenuse in various right triangle problems.

  3. Analyze right triangle scenarios -

    Identify which side is the hypotenuse and determine the correct approach for each geometry question.

  4. Evaluate your quiz performance -

    Leverage instant feedback to spot errors, understand misconceptions, and reinforce correct methods.

  5. Improve problem-solving speed and accuracy -

    Sharpen calculation skills under timed conditions to boost confidence and efficiency in geometry tasks.

  6. Develop long-term geometry confidence -

    Track your progress across multiple questions and build lasting mastery of right triangle concepts.

Cheat Sheet

  1. Fundamental Formula: a² + b² = c² -

    Every right triangle follows this core relationship, where the square of the hypotenuse (c) equals the sum of the squares of the legs (a and b). For example, in a classic 3-4-5 triangle, 3² + 4² = 5² (9 + 16 = 25), a fact you'll confirm in any Pythagorean Theorem Quiz item.

  2. Distinguishing Legs vs. Hypotenuse -

    In right triangle math quizzes, correctly identifying the two legs (the sides forming the right angle) and the hypotenuse (opposite the right angle) is crucial. Remember: the hypotenuse is always the longest side, so start there when you see a Pythagorean theorem practice quiz question.

  3. Converse Theorem for Missing Sides -

    The Converse of the Pythagorean Theorem states that if a² + b² = c², the triangle is right-angled, a handy trick during a Pythagorean theorem test. Use this to check if a set of lengths like 6, 8, and 10 form a right triangle (6² + 8² = 10²).

  4. Common Pythagorean Triples -

    Memorizing triples such as 3-4-5, 5-12-13, and 8-15-17 speeds you through a geometry quiz online without needing a calculator. These integer sets pop up often in both straightforward and word-based Pythagorean theorem practice quiz problems.

  5. Word Problems & Real-World Applications -

    Translating scenarios - like ladder height or distance across a field - into Pythagorean equations builds your problem-solving muscle for any right triangle math quiz. Sketch a diagram, label sides a, b, and c, then apply a² + b² = c² for instant clarity.

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