Question

How does the
Pythagorean Theorem relate the side lengths of
a right triangle?
2. Use Structure A side of each of the
three squares forms a side of
a right triangle.
Would any three squares
form the sides of a right
triangle? Explain.
3. Construct Arguments Xavier said the missing
length is about 18.5 units. Without calculating,
how can you tell that Xavier solved
incorrectly?
Do You Know How?
4. A right triangle has leg lengths of 4 inches and
5 inches. What is the length of the hypotenuse?
Write the answer as a square root and round to
the nearest tenth of an inch.
5. Find the missing side
length to the nearest
tenth of a foot.
6. Find the missing side
length to the neares

Answer 1

1) Pythagorean Theorem
- In a right triangle with legs a and b and hypotenuse c, the theorem says a^2 + b^2 = c^2.
- In words: the sum of the squares of the two legs equals the square of the hypotenuse.

2) Use Structure — three squares forming the sides of a right triangle
- If the side lengths of the three squares are s1, s2, s3 and those sides are the three sides of a triangle, then for the triangle to be right with s3 as the hypotenuse you must have
s1^2 + s2^2 = s3^2.
- Note s1^2, s2^2, s3^2 are the areas of the three squares. So two of the square areas must add to the third square’s area.
- Conclusion: No — not any three squares will do. Only triples whose side lengths satisfy the Pythagorean relation (equivalently, whose areas satisfy area1 + area2 = area3) can form a right triangle. Example: side lengths 3, 4, 5 (areas 9, 16, 25) work because 9 + 16 = 25.

3) Construct arguments — Xavier’s 18.5 is incorrect (how to tell without full calculation)
- General quick checks you can do without full calculation:
- The hypotenuse must be longer than each leg but shorter than the sum of the two legs (triangle inequality). If 18.5 is less than the longest given leg or greater than the sum of the two given legs, it must be wrong.
- Square or estimate roughly: compare 18.5^2 (≈342.25) to the sum of the squares of the legs. If 18.5^2 is much larger or much smaller than that sum, 18.5 is wrong.
- Compare to nearby known triples or estimates: for example, if the legs are around 10 and 12, the hypotenuse is about sqrt(100+144)=sqrt244≈15.6, so 18.5 would be clearly too large.
- If you give the specific leg lengths/diagram Xavier used, I’ll show the quick check applied to that case.

4) Right triangle with legs 4 in and 5 in
- Hypotenuse = sqrt(4^2 + 5^2) = sqrt(16 + 25) = sqrt(41).
- Rounded to the nearest tenth: sqrt(41) ≈ 6.4 inches.

5) and 6) The problems are cut off in your message. Please send the missing side lengths or the diagram for problems 5 and 6 and I’ll solve them to the nearest tenth.