Are The Triangles Below Right Triangles? If Not, What Kind Of Triangles Are They? 4195
- Determine Whether The Numbers Can Represent The Side Lengths Of A Triangle. If They Can, Classify The Triangle As Acute, Right, Or Obtuse. Show Work!
- 26, 35, 62
- 29, 18, 14
- 30, 78, 72
- 17, 19, 22
The Correct Answer and Explanation is:
Let’s evaluate each triangle using the Pythagorean Theorem and triangle inequality.
Part 1: Are the triangles right triangles?
Problem 8: Sides = 9, 12, 15
Check if 92+122=1529^2 + 12^2 = 15^2:
81+144=22581 + 144 = 225
225=225225 = 225 ✅
Answer: Right triangle
Problem 9: Sides = 15, 36, 4954\sqrt{95}
Calculate 495≈4×9.75≈394\sqrt{95} ≈ 4 × 9.75 ≈ 39. Check:
Is 152+362=(495)215^2 + 36^2 = (4\sqrt{95})^2?
Left: 225+1296=1521225 + 1296 = 1521
Right: (495)2=16×95=1520(4\sqrt{95})^2 = 16 × 95 = 1520
1521≠15201521 ≠ 1520 ❌
Since hypotenuse² < sum of squares of legs ⇒ Acute triangle
Problem 10: Sides = 7, 8, 113\sqrt{113}
113≈10.63\sqrt{113} ≈ 10.63
Check if 72+82=(113)27^2 + 8^2 = (\sqrt{113})^2:
49+64=11349 + 64 = 113, and (113)2=113(\sqrt{113})^2 = 113 ✅
Answer: Right triangle
Part 2: Can the numbers form a triangle? What kind?
Use Triangle Inequality: The sum of the two shorter sides must be greater than the longest.
Problem 11: 26, 35, 62
26 + 35 = 61 < 62 ❌
Answer: Not a triangle
Problem 12: 14, 18, 29
14 + 18 = 32 > 29 ✅
14² + 18² = 196 + 324 = 520
29² = 841
Since 520 < 841 ⇒ Obtuse triangle
Problem 13: 30, 72, 78
30 + 72 = 102 > 78 ✅
30² + 72² = 900 + 5184 = 6084
78² = 6084
6084 = 6084 ✅
Answer: Right triangle
Problem 14: 17, 19, 22
17 + 19 = 36 > 22 ✅
17² + 19² = 289 + 361 = 650
22² = 484
650 > 484 ⇒ Acute triangle
Summary:
| Problem | Triangle Type |
|---|---|
| 8 | Right |
| 9 | Acute |
| 10 | Right |
| 11 | Not a triangle |
| 12 | Obtuse |
| 13 | Right |
| 14 | Acute |
Explanation
To determine the type of triangle, we apply two main methods: the Pythagorean Theorem and the Triangle Inequality Theorem. The Pythagorean Theorem states that for a triangle to be right-angled, the square of the longest side (hypotenuse) must equal the sum of the squares of the other two sides. If it’s greater, the triangle is obtuse; if it’s less, the triangle is acute.
In Problems 8 and 10, the side lengths satisfy the Pythagorean Theorem exactly, confirming that these are right triangles. For Problem 9, the calculated hypotenuse is slightly less than the sum of squares of the legs, meaning it is an acute triangle, as all angles are less than 90°.
For Problems 11–14, we first verify if the side lengths can form a triangle using the Triangle Inequality Theorem, which states the sum of any two sides must be greater than the third side. In Problem 11, this is not true, so the set cannot form a triangle.
For Problems 12 to 14, we then check the relationship between the square of the longest side and the sum of squares of the two shorter sides. If the longest side squared is greater, it’s obtuse (like in Problem 12). If it’s equal, the triangle is right (Problem 13). If it’s less, it’s acute (Problem 14).
These concepts help classify triangles by their angles using only side lengths, which is crucial in geometry and trigonometry.
