Homework 1 Pythagorean Theorem And Its Converse Answers

The Pythagorean Theorem and Its Converse are staple topics in any introductory geometry curriculum, and Homework 1 often asks students to apply both statements to solve a variety of problems. Understanding not only the formulas but also the logical structure behind the theorem and its converse is essential for mastering the material and achieving full marks on the assignment. This article breaks down the concepts, provides step‑by‑step solution methods, presents common homework questions with complete answers, and offers tips for checking work—everything you need to ace Homework 1 on the Pythagorean Theorem and its converse.

Quick note before moving on.

Introduction: Why the Pythagorean Theorem Matters

The classic statement of the Pythagorean Theorem is:

In a right‑angled triangle, the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

Mathematically, if a right triangle has legs (a) and (b) and hypotenuse (c), then

[ c^{2}=a^{2}+b^{2}. ]

The converse flips the logical direction:

If the squares of two sides of a triangle sum to the square of the third side, then the triangle is right‑angled.

In symbols, if (a^{2}+b^{2}=c^{2}) (with (c) the longest side), then the angle opposite side (c) is (90^{\circ}).

Both statements are used repeatedly in homework problems: the theorem to find missing lengths in a known right triangle, and the converse to test whether a given set of side lengths can form a right triangle. Mastery of these ideas unlocks a wide range of geometry tasks, from simple distance calculations to more complex applications in coordinate geometry and trigonometry.

1. How to Use the Pythagorean Theorem

1.1 Identify the Right Triangle

1. Look for a 90° angle explicitly marked in the diagram, or
2. Determine the longest side; if the problem asks you to verify right‑angledness, you’ll use the converse first.

1.2 Choose the Correct Formula

• Find the hypotenuse: (c = \sqrt{a^{2}+b^{2}})
• Find a leg: (a = \sqrt{c^{2}-b^{2}}) (or (b = \sqrt{c^{2}-a^{2}}))

1.3 Solve Step‑by‑Step

1. Write down the known values and assign letters.
2. Plug them into the appropriate equation.
3. Isolate the unknown and take the square root.
4. Check the answer by substituting back into the original equation.

1.4 Example Problem (Homework 1, Question 1)

Given a right triangle with legs 6 cm and 8 cm, find the hypotenuse.

Solution

[ c = \sqrt{6^{2}+8^{2}} = \sqrt{36+64} = \sqrt{100}=10\text{ cm}. ]

The hypotenuse is 10 cm.

2. How to Use the Converse of the Pythagorean Theorem

2.1 When to Apply the Converse

• Testing side lengths: You have three numbers and need to know if they can be the sides of a right triangle.
• Proving right angles: In coordinate geometry, you can compute distances between points; if the distance relationship satisfies the converse, the angle is right.

2.2 Procedure

1. Arrange the numbers so the largest is assumed to be the hypotenuse (c).
2. Square each number and compute (a^{2}+b^{2}).
3. Compare (a^{2}+b^{2}) with (c^{2}).
   • If they are equal, the triangle is right‑angled.
   • If not, the triangle is not right‑angled.

2.3 Example Problem (Homework 1, Question 2)

Do the side lengths 5 cm, 12 cm, and 13 cm form a right triangle?

Solution

Arrange: (c=13), (a=5), (b=12).

[ a^{2}+b^{2}=5^{2}+12^{2}=25+144=169,\qquad c^{2}=13^{2}=169. ]

Since (a^{2}+b^{2}=c^{2}), the triangle is right‑angled.

3. Common Homework Scenarios and Full Answers

Below are typical Homework 1 questions, each followed by a detailed solution. The same reasoning can be adapted to similar problems you may encounter.

3.1 Finding Missing Legs

Question 3: A right triangle has a hypotenuse of 15 cm and one leg of 9 cm. Find the length of the other leg.

Answer:

[ \begin{aligned} c^{2} &= a^{2}+b^{2}\ 15^{2} &= 9^{2}+b^{2}\ 225 &= 81+b^{2}\ b^{2} &= 144\ b &= \sqrt{144}=12\text{ cm}. \end{aligned} ]

The missing leg is 12 cm.

3.2 Determining Whether a Triangle Is Right‑Angled

Question 4: Are the side lengths 7, 24, and 25 the sides of a right triangle?

Answer:

Largest side (c=25) Not complicated — just consistent..

[ 7^{2}+24^{2}=49+576=625,\qquad 25^{2}=625. ]

Equality holds, so yes, the triangle is right‑angled.

3.3 Distance Between Two Points (Coordinate Geometry)

Question 5: Points (A(2,3)) and (B(10,7)) are endpoints of a segment. Is the segment the hypotenuse of a right triangle with the third vertex at the origin (O(0,0))?

Answer:

Compute distances:

[ OA = \sqrt{2^{2}+3^{2}} = \sqrt{4+9}= \sqrt{13}, ]

[ OB = \sqrt{10^{2}+7^{2}} = \sqrt{100+49}= \sqrt{149}, ]

[ AB = \sqrt{(10-2)^{2}+(7-3)^{2}} = \sqrt{8^{2}+4^{2}} = \sqrt{64+16}= \sqrt{80}. ]

Now test the converse with the longest side (OB):

[ OA^{2}+AB^{2}=13+80=93\neq149=OB^{2}. ]

Thus the triangle is not right‑angled at (O). Even so, if we test with (AB) as the longest side (which it is not), the equality still fails, confirming the triangle is not right‑angled Not complicated — just consistent..

3.4 Real‑World Application: Ladder Problem

Question 6: A 13‑ft ladder leans against a wall, touching the ground 5 ft away from the wall. How high up the wall does the ladder reach?

Answer:

Treat the ladder as the hypotenuse (c=13) ft, the distance from the wall as one leg (a=5) ft, and the height as the other leg (b) Small thing, real impact. Worth knowing..

[ b = \sqrt{c^{2}-a^{2}} = \sqrt{13^{2}-5^{2}} = \sqrt{169-25}= \sqrt{144}=12\text{ ft}. ]

The ladder reaches 12 ft up the wall.

3.5 Verifying a Triangle from Word Problems

Question 7: A rectangular garden measures 9 m by 40 m. A diagonal walkway runs from one corner to the opposite corner. What is the length of the walkway, and does it satisfy the Pythagorean Theorem?

Answer:

The garden forms a right rectangle; the diagonal is the hypotenuse.

[ d = \sqrt{9^{2}+40^{2}} = \sqrt{81+1600}= \sqrt{1681}=41\text{ m}. ]

Since (9^{2}+40^{2}=41^{2}), the diagonal does satisfy the theorem, confirming the rectangle’s right angles.

4. Frequently Asked Questions (FAQ)

4.1 Can the Pythagorean Theorem be used for non‑right triangles?

No. And the theorem only holds for right‑angled triangles. For non‑right triangles, you must use the Law of Cosines, which reduces to the Pythagorean Theorem when the included angle is (90^{\circ}) Practical, not theoretical..

4.2 What if the side lengths are not integers?

The theorem works for any real numbers. Square the lengths, add or subtract as required, and take the square root. Decimal answers are perfectly acceptable Less friction, more output..

4.3 Is the converse always true?

Yes. Also, the converse of the Pythagorean Theorem is a proven theorem itself: if (a^{2}+b^{2}=c^{2}) for three positive lengths with (c) the longest, the triangle must be right‑angled. This is often used to prove right angles in geometry proofs Most people skip this — try not to..

4.4 How do I avoid sign errors when solving for a leg?

Always square the known sides first, then subtract the smaller squared value from the larger squared value before taking the square root. The result will be non‑negative, representing a length.

4.5 Can I use the theorem in three‑dimensional problems?

In 3‑D, the distance formula between two points ((x_{1},y_{1},z_{1})) and ((x_{2},y_{2},z_{2})) is an extension:

[ d = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}. ]

While not a direct application of the 2‑D Pythagorean Theorem, the same principle (sum of squares) underlies the calculation Simple as that..

5. Tips for Completing Homework 1 Efficiently

1. Label the diagram clearly; assign (a), (b), and (c) before plugging numbers.
2. Check which side is longest before deciding which variable is the hypotenuse.
3. Write the equation first, then isolate the unknown—this prevents algebraic mistakes.
4. Double‑check by squaring the answer and confirming it satisfies the original equation.
5. Use a calculator for square roots only when the result is not a perfect square; otherwise, simplify to an integer for a cleaner answer.
6. Show all work even if the problem seems simple; teachers often award partial credit for clear reasoning.

Conclusion

Homework 1 on the Pythagorean Theorem and its converse tests both computational skill and logical reasoning. By mastering the identification of right triangles, correctly applying the theorem to find missing lengths, and confidently using the converse to verify right angles, you’ll solve every problem the assignment presents. Now, remember to follow the systematic steps—identify the longest side, square the known lengths, perform the addition or subtraction, and finally take the square root—while always checking your work. With these strategies and the sample solutions above, you are well equipped to achieve full marks and deepen your geometric understanding No workaround needed..

Worth pausing on this one.