Identifying Lengths of the Sides of a Right Triangle. Which are the side lengths of a right triangle? Check all that apply. 3, 14, and square root 205 6,11, and square root 158 19, 180, and 181 3, 19, and square root 380 2, 9, and square root 85
The Correct Answer and Explanation is:
To determine which of the given sets of numbers can represent the side lengths of a right triangle, we use the Pythagorean Theorem, which states:a2+b2=c2a^2 + b^2 = c^2a2+b2=c2
Here, aaa and bbb are the legs of the triangle (shorter sides), and ccc is the hypotenuse (the longest side). We need to check whether the sum of the squares of the two shorter sides equals the square of the longest side.
Let’s check each set:
1. 3, 14, and 205\sqrt{205}205
Find squares of the legs:32=9,142=196,2052=2053^2 = 9,\quad 14^2 = 196,\quad \sqrt{205}^2 = 205 32=9,142=196,2052=2059+196=205⇒This is a right triangle. ✅9 + 196 = 205 \Rightarrow \text{This is a right triangle. ✅}9+196=205⇒This is a right triangle. ✅
2. 6, 11, and 158\sqrt{158}15862=36,112=121,1582=1586^2 = 36,\quad 11^2 = 121,\quad \sqrt{158}^2 = 158 62=36,112=121,1582=15836+121=157≠158⇒Not a right triangle. ❌36 + 121 = 157 \neq 158 \Rightarrow \text{Not a right triangle. ❌}36+121=157=158⇒Not a right triangle. ❌
3. 19, 180, and 181192=361,1802=32,400,1812=32,76119^2 = 361,\quad 180^2 = 32{,}400,\quad 181^2 = 32{,}761 192=361,1802=32,400,1812=32,761361+32,400=32,761⇒This is a right triangle. ✅361 + 32{,}400 = 32{,}761 \Rightarrow \text{This is a right triangle. ✅}361+32,400=32,761⇒This is a right triangle. ✅
4. 3, 19, and 380\sqrt{380}38032=9,192=361,3802=3803^2 = 9,\quad 19^2 = 361,\quad \sqrt{380}^2 = 380 32=9,192=361,3802=3809+361=370≠380⇒Not a right triangle. ❌9 + 361 = 370 \neq 380 \Rightarrow \text{Not a right triangle. ❌}9+361=370=380⇒Not a right triangle. ❌
5. 2, 9, and 85\sqrt{85}8522=4,92=81,852=852^2 = 4,\quad 9^2 = 81,\quad \sqrt{85}^2 = 85 22=4,92=81,852=854+81=85⇒This is a right triangle. ✅4 + 81 = 85 \Rightarrow \text{This is a right triangle. ✅}4+81=85⇒This is a right triangle. ✅
Correct Answers:
✅ 3, 14, and 205\sqrt{205}205
✅ 19, 180, and 181
✅ 2, 9, and 85\sqrt{85}85
Explanation (
To determine if three given lengths can form the sides of a right triangle, we use the Pythagorean Theorem, one of the most important principles in geometry. This theorem states that for a triangle to be right-angled, the square of the longest side (called the hypotenuse) must be equal to the sum of the squares of the other two sides (called the legs). Mathematically, if the side lengths are aaa, bbb, and ccc, with ccc being the longest side, then the triangle is a right triangle if and only if:a2+b2=c2a^2 + b^2 = c^2a2+b2=c2
This test helps us verify whether a triangle with given side lengths satisfies the conditions for being a right triangle. When square roots are involved, we square them to eliminate the radical sign and make comparison easier.
In this case, we tested all five sets using the theorem. The sets {3, 14, 205\sqrt{205}205}, {19, 180, 181}, and {2, 9, 85\sqrt{85}85} satisfied the condition. That means the sum of the squares of the two smaller sides was exactly equal to the square of the longest side, confirming they can form right triangles.
The other sets failed this test because the sums of the squares of the legs did not equal the square of the hypotenuse, so they cannot form right triangles.
Using the Pythagorean Theorem to test side lengths is a reliable and straightforward method in both pure mathematics and practical applications like construction and design.
