Constructing Triangles

Free sample questions, a clear explanation, and 5 practice skills with an AI tutor that guides without giving the answer away.

Concept Review

The Triangle Inequality: Why Some Triangles Are Impossible

Imagine you're building a triangular picture frame with three wooden sticks. You have sticks measuring 3 inches, 4 inches, and 10 inches. No matter how hard you try, you cannot connect the ends to form a triangle. But why?

The answer lies in one of geometry's most fundamental rules: the Triangle Inequality Theorem. This theorem tells us that for any triangle to exist, the sum of any two sides must always be greater than the third side.

Testing the Rule

Let's check our 3-4-10 triangle attempt:

✗3 + 4 = 7, which is less than 10
✓3 + 10 = 13, which is greater than 4
✓4 + 10 = 14, which is greater than 3

Since one test failed, this triangle is impossible. Think of it this way: if you place the 3-inch and 4-inch sticks end-to-end, you only get 7 inches total—not nearly enough to reach the other end of the 10-inch stick!

🔑 Key Insight

The two shorter sides of any triangle must work together to "reach" the longest side. If they're too short, even when combined, there's literally no way to close the triangle. It's like trying to build a bridge that's too short to span a river.

A Triangle That Works

Now let's try sides of 5, 7, and 9 units:

✓5 + 7 = 12, which is greater than 9
✓5 + 9 = 14, which is greater than 7
✓7 + 9 = 16, which is greater than 5

All three tests pass! This triangle can definitely be built. The shortest two sides (5 and 7) add up to 12, which easily "reaches across" the longest side of 9.

🔑 Key Takeaway

Just like our wooden picture frame, every triangle in the real world—from the supports in bridges to the slices of pizza—must follow the Triangle Inequality Theorem. Before architects, engineers, or even pizza makers create triangular shapes, the math has already determined what's possible and what's not. The numbers don't lie: some triangles simply cannot exist, no matter how much we want them to!

Sample questions

1. Can sides of length 4, 5, and 6 form a triangle?

○ Yes, because 4 + 5 > 6, 4 + 6 > 5, and 5 + 6 > 4

○ No, because 4 + 5 = 9 which is greater than 6, so it should work

✓ Both A and C

○ Yes, all sums are greater than the third side

Answer: Both A and C — Triangle Inequality Theorem: sum of any two sides must be greater than the third. All conditions are satisfied.

2. Can sides of length 2, 3, and 6 form a triangle?

✓ All of the above are correct reasons

○ Yes, because 2 + 3 = 5, which is less than 6, so no

○ No, because 2 + 3 < 6

○ No, the sum of the two smaller sides must be greater than the largest

Answer: All of the above are correct reasons — 2 + 3 = 5 < 6, so it cannot form a triangle.

3. Which set of side lengths could form a triangle?

○ 3, 4, 7

○ 3, 4, 8

○ 3, 4, 9

✓ 3, 4, 6

Answer: 3, 4, 6 — 3 + 4 = 7, which is not > 7 (equal doesn't count). 3+4=7 < 8 and <9, so only 3+4=7 > 6 works.

Skills in this topic

Determine if three given side lengths can form a triangle (Triangle Inequality Theorem)
Determine if three given angle measures can form a triangle
Draw a triangle given three side lengths using a ruler and compass
Identify conditions that result in a unique triangle, more than one triangle, or no triangle
Understand the relationship between side lengths and their opposite angle measures

Practice 50+ questions on this topic

Unlimited interactive practice, progress tracking, and Nova — your AI tutor. Free to start.

Start learning free →