Converse of Pythagorean Theorem Worksheet - Math Monks - Free Printable

Explanation:
We are using the Converse of the Pythagorean Theorem, which says:

> If the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is a right triangle.

So for each triangle (or set of three numbers), we:
1. Identify the longest side (hypotenuse candidate).
2. Square all three sides.
3. Check if:
 (longest side)² = (side₁)² + (side₂)²
If yes → “Yes” and mark the right angle (usually between the two shorter sides).
If no → “No”.

Let’s go one by one.

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1) Sides: 9, 12, 15
Longest = 15
Check:
9² = 81
12² = 144
15² = 225
81 + 144 = 225 ✔
→ Yes. Right angle between sides 9 and 12.

2) Sides: 18, 24, 30
Longest = 30
18² = 324
24² = 576
30² = 900
324 + 576 = 900 ✔
→ Yes. Right angle between 18 and 24.

3) Sides: 7, 13, 17
Longest = 17
7² = 49
13² = 169
17² = 289
49 + 169 = 218 ≠ 289 ✘
→ No.

4) Sides: 8, 7, √113
First, compute (√113)² = 113
8² = 64
7² = 49
64 + 49 = 113 ✔
→ Yes. Right angle between 8 and 7.

5) Sides: 3.9, 10.3, 11
Longest = 11
3.9² = 15.21
10.3² = 106.09
11² = 121
15.21 + 106.09 = 121.30 ≈ 121.3 ≠ 121
But wait — let’s check more precisely:
3.9 = 39/10 → (39/10)² = 1521/100 = 15.21
10.3 = 103/10 → (103/10)² = 10609/100 = 106.09
Sum = (1521 + 10609)/100 = 12130/100 = 121.30
11² = 121
121.30 ≠ 121 → Not equal
→ No.

6) Sides: 4, 8, 8.9
Longest = 8.9
4² = 16
8² = 64
8.9² = ?
8.9 × 8.9 = (9 − 0.1)² = 81 − 1.8 + 0.01 = 79.21
16 + 64 = 80 ≠ 79.21
→ No.

Now the second part: “Do the following 3 sides make a right triangle?”

We do same check.

a) 10, 15, 5√13
First, find (5√13)² = 25 × 13 = 325
10² = 100
15² = 225
100 + 225 = 325 ✔
→ Yes.

b) 6, 2, 2√10
(2√10)² = 4 × 10 = 40
6² = 36
2² = 4
36 + 4 = 40 ✔
→ Yes.

c) 2√14, 13, 15
Find squares:
(2√14)² = 4 × 14 = 56
13² = 169
15² = 225
Which is largest? 225 → 15 is longest.
Check: 56 + 169 = 225 ✔
→ Yes.

d) 5, 10, 14
Largest = 14
5² = 25
10² = 100
14² = 196
25 + 100 = 125 ≠ 196 ✘
→ No.

e) 6, 5, √61
(√61)² = 61
6² = 36
5² = 25
36 + 25 = 61 ✔
→ Yes.

f) √3, 2, 3
Squares:
(√3)² = 3
2² = 4
3² = 9
3 + 4 = 7 ≠ 9 ✘
→ No.

Now compile all answers in order:

Triangles 1–6:
1) Yes
2) Yes
3) No
4) Yes
5) No
6) No

Triples:
10, 15, 5√13 → Yes
6, 2, 2√10 → Yes
2√14, 13, 15 → Yes
5, 10, 14 → No
6, 5, √61 → Yes
√3, 2, 3 → No

Final Answer:
1) Yes
2) Yes
3) No
4) Yes
5) No
6) No
10, 15, 5√13: Yes
6, 2, 2√10: Yes
2√14, 13, 15: Yes
5, 10, 14: No
6, 5, √61: Yes
√3, 2, 3: No