Question:** A triangle has sides of length 7 cm, 10 cm, and 5 cm. Determine if it is a right triangle.
["Determine If a Triangle with Sides 7 cm, 10 cm, and 5 cm Is a Right Triangle", "When learning geometry, one of the foundational skills is identifying right triangles. A key criterion is the Pythagorean Theorem, which states that in a right triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. This principle allows us to quickly determine if a triangle with given side lengths — in this case, 7 cm, 10 cm, and 5 cm — qualifies as a right triangle.", "---", "### Step 1: Identify the Longest Side (Hypotenuse Candidate)", "Among the three side lengths:", "- 10 cm\n- 7 cm\n- 5 cm", "The longest side is 10 cm, which would be the hypotenuse if this triangle is right-angled.", "---", "### Step 2: Apply the Pythagorean Theorem", "The Pythagorean Theorem formula is:\n[\nc^2 = a^2 + b^2\n]\nwhere ( c ) is the hypotenuse, and ( a ) and ( b ) are the other two sides.", "Substituting the values:\n[\n10^2 = 7^2 + 5^2\n]\n[\n100 = 49 + 25\n]\n[\n100 = 74\n]", "Since ( 100 <br/>\neq 74 ), the equation does not hold true.", "---", "### Step 3: Analyze the Result", "Because the square of the longest side is not equal to the sum of the squares of the other two sides, the triangle is not a right triangle.", "---", "### Alternative Check: Triangle Inequality Theorem", "Even before applying the Pythagorean Theorem, it's prudent to confirm whether a triangle can exist with these side lengths. The Triangle Inequality Theorem states that the sum of any two sides must be greater than the third side.", "- ( 7 + 5 = 12 > 10 )\n- ( 7 + 10 = 17 > 5 )\n- ( 5 + 10 = 15 > 7 )", "All conditions are satisfied, so the sides can form a triangle—but not necessarily a right one.", "---", "### Conclusion", "To wrap up:\n✅ The triangle has valid side lengths that satisfy the triangle inequality.\n❌ However, since ( 10^2 <br/>\neq 7^2 + 5^2 ), it is not a right triangle.", "Understanding the Pythagorean Theorem and applying it systematically helps you quickly verify right triangles in geometry problems—this is a key skill for students, teachers, and geometry enthusiasts alike.", "If you're checking triangle types, always start with the hypothesis (right triangle), test the hypothesis using the longest side, and then draw a conclusion.", "---", "Keywords for SEO:\nright triangle test, Pythagorean Theorem, determine right triangle, triangle side lengths, 7 cm 10 cm 5 cm triangle, geometric proofs, math learners, triangle classification, right triangle criteria, geometry formula, check triangle validity", "Related reads:\nHow to Test if a Triangle Is Right, Using the Pythagorean Theorem, Triangle Types and Properties, Geometry Practice Problems", "---", "Understanding these fundamental concepts boosts your geometry skills and helps in solving more advanced math problems with confidence."]
