Question

1. the pythagorean theorem can be proven using areas of squares built on each side of a right triangle. describe how the areas of these squares demonstrate that (a^2 + b^2 = c^2).

4.2 converse of pt

1. determine whether the triangle with side lengths 6 cm, 8 cm, and 10 cm is a right triangle. show your work using the pythagorean theorem or its converse.
2. a triangle has side lengths 7 m, 24 m, and 25 m. use the converse of the pythagorean theorem to determine if the triangle is a right triangle.
3. a right triangle has one leg measuring 9 units and a hypotenuse measuring 15 units. find the length of the other leg using the pythagorean theorem.
4. a ladder is leaning against a wall. the bottom of the ladder is 9 ft from the wall, and the top touches the wall.

• use the pythagorean theorem to determine if a 15 ft ladder would reach the wall exactly.
• if not, explain whether the ladder would be too short or too long, and how you know.

Explanation:

Problem 1: Determine if triangle with sides 6 cm, 8 cm, 10 cm is right triangle

Step1: Recall Converse of Pythagorean Theorem

The converse states that if \(a^2 + b^2 = c^2\) (where \(c\) is the longest side), then the triangle is right - angled. Here, \(c = 10\) cm, \(a = 6\) cm, \(b = 8\) cm.

Step2: Calculate \(a^{2}+b^{2}\) and \(c^{2}\)

Calculate \(a^{2}=6^{2}=36\), \(b^{2}=8^{2} = 64\). Then \(a^{2}+b^{2}=36 + 64=100\).
Calculate \(c^{2}=10^{2}=100\).
Since \(a^{2}+b^{2}=c^{2}\) (100 = 100), by the converse of the Pythagorean Theorem, the triangle is a right triangle.

Problem 2: Determine if triangle with sides 7 m, 24 m, 25 m is right triangle

Step1: Recall Converse of Pythagorean Theorem

For a triangle with sides \(a\), \(b\), \(c\) ( \(c\) is the longest side), if \(a^{2}+b^{2}=c^{2}\), it is right - angled. Here, \(c = 25\) m, \(a = 7\) m, \(b = 24\) m.

Step2: Calculate \(a^{2}+b^{2}\) and \(c^{2}\)

\(a^{2}=7^{2}=49\), \(b^{2}=24^{2}=576\). Then \(a^{2}+b^{2}=49 + 576 = 625\).
\(c^{2}=25^{2}=625\).
Since \(a^{2}+b^{2}=c^{2}\) (625 = 625), by the converse of the Pythagorean Theorem, the triangle is a right triangle.

Problem 3: Find the length of the other leg of a right triangle with one leg 9 units and hypotenuse 15 units

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse and \(a\), \(b\) are the legs. Let the unknown leg be \(x\), one leg \(a = 9\), hypotenuse \(c = 15\). So, \(9^{2}+x^{2}=15^{2}\).

Step2: Solve for \(x\)

\(81+x^{2}=225\). Subtract 81 from both sides: \(x^{2}=225 - 81=144\). Take the square root of both sides: \(x=\sqrt{144} = 12\) units.

Problem 4: Ladder leaning against a wall (assuming the height of the wall from the ground to the top of the ladder's touch - point is what we need to find, and the wall is vertical, so it's a right triangle with base 9 ft, hypotenuse 15 ft)

Answer:

s:

1. The triangle with sides 6 cm, 8 cm, 10 cm is a right triangle.
2. The triangle with sides 7 m, 24 m, 25 m is a right triangle.
3. The length of the other leg is 12 units.
4. The height the 15 - ft ladder reaches on the wall (when the base is 9 ft from the wall) is 12 ft. If the wall's height is 12 ft, the ladder reaches exactly.