Q1. Find the value of x. To start, use the Pythagorean Theorem. Then substitute 9 for a, 12 for b, and x for c.

Background

Topic: The Pythagorean Theorem

This question tests your ability to apply the Pythagorean Theorem to solve for the length of the hypotenuse in a right triangle.

Key Terms and Formula:

• Pythagorean Theorem:

• Where and are the legs of the right triangle, and is the hypotenuse.

Step-by-Step Guidance

1. Identify the values: , , .

2. Write the Pythagorean Theorem: .

3. Substitute the given values: .

4. Calculate and separately, then add them together.

5. Set the sum equal to and prepare to solve for by taking the square root of both sides.

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Q2. Does each set of numbers form a Pythagorean triple? (a) 6, 8, 10 (b) 7, 16, 18

Background

Topic: Pythagorean Triples

This question asks you to determine if the given sets of numbers can be the side lengths of a right triangle, meaning they satisfy the Pythagorean Theorem.

Key Terms and Formula:

• Pythagorean Triple: Three positive integers , , such that .

Step-by-Step Guidance

1. For each set, identify the largest number as (the hypotenuse), and the other two as and .

2. Calculate for each set.

3. Calculate for each set.

4. Compare to to see if they are equal.

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Q3. A window washer has an 18-ft ladder. He needs to reach the bottom of a window 16 feet off the ground. How far out from the building should the base of the ladder be? Round to the nearest tenth of a foot.

Background

Topic: Right Triangle Applications (Pythagorean Theorem)

This problem involves applying the Pythagorean Theorem to a real-world scenario involving a ladder, a wall, and the ground.

Key Terms and Formula:

• Pythagorean Theorem:

• Here, is the length of the ladder (hypotenuse), is the height reached (16 ft), and is the distance from the wall (what you are solving for).

Step-by-Step Guidance

1. Assign the values: ft (vertical), ft (ladder), = distance from wall (unknown).

2. Write the Pythagorean Theorem: .

3. Substitute the known values: .

4. Calculate and .

5. Rearrange to solve for by subtracting from .

6. Prepare to take the square root to find and round to the nearest tenth.

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Q4. A square has a diagonal of 12 cm. What is the perimeter of the square? Express in simplest radical form.

Background

Topic: Properties of Squares and the Pythagorean Theorem

This question tests your understanding of the relationship between the side length and diagonal of a square, using the Pythagorean Theorem.

Key Terms and Formula:

• For a square with side length , the diagonal is .

• Perimeter of a square: .

Step-by-Step Guidance

1. Set up the equation for the diagonal: , where cm.

2. Solve for by dividing both sides by .

3. Express in simplest radical form.

4. Use to write the perimeter in terms of .

5. Substitute your expression for into the perimeter formula.

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Q5. The lengths of the sides of a triangle are given. Classify each triangle as acute, right, or obtuse. (a) 3, 4, 6 (b) 9, 11, 16 (c) 4, 6, 7

Background

Topic: Triangle Classification by Side Lengths

This question asks you to use the converse of the Pythagorean Theorem to classify triangles as acute, right, or obtuse based on their side lengths.

Key Terms and Formula:

• Let be the longest side, and the other sides.

• If , the triangle is right.

• If , the triangle is acute.

• If , the triangle is obtuse.

Step-by-Step Guidance

1. For each set, identify the largest number as .

2. Calculate and for each set.

3. Compare to to determine the triangle type.

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Q6. Open-Ended: Find integers j and k such that (a) the two given integers and j represent the side lengths of an acute triangle, and (b) the two given integers and k represent the side lengths of an obtuse triangle. (Given: 33, 55)

Background

Topic: Triangle Inequality and Classification

This question asks you to find possible integer values for a third side so that the triangle formed is acute or obtuse, using the triangle inequality and the converse of the Pythagorean Theorem.

Key Terms and Formula:

• Triangle Inequality: The sum of any two sides must be greater than the third side.

• For an acute triangle:

• For an obtuse triangle:

Step-by-Step Guidance

1. Let the three sides be 33, 55, and (or ).

2. Apply the triangle inequality to find the possible range for and .

3. For the acute case, set up and solve for .

4. For the obtuse case, set up and solve for .

Try solving on your own before revealing the answer!