TEKS 8.7C Worksheets — Grade 8 Use the Pythagorean Theorem and its converse

Santiago is helping his school organize a fundraiser in Austin, Texas. They plan to set up a large rectangular booth in the park that is 12 feet wide and 16 feet long. To make sure the booth is stable, they want to secure a ladder that reaches from the ground to the top corner of the booth. How long does the ladder need to be if it is placed against one of the side corners of the booth? Use the Pythagorean Theorem to find your answer.

1. 20 feet ✓
2. 16 feet
3. 12 feet
4. 15 feet

Why: To find the length of the ladder, we can use the Pythagorean Theorem, which states that a² + b² = c², where c is the hypotenuse (the length of the ladder) and a and b are the other two sides of the right triangle formed by the booth. In this case, a = 12 feet and b = 16 feet. So we calculate: 12² + 16² = c². This gives us 144 + 256 = c², which simplifies to 400 = c². Taking the square root of both sides gives us c = 20 feet. Therefore, the ladder needs to be 20 feet long.

Lucy is helping her father build a new deck in their backyard in El Paso. The deck will be in the shape of a right triangle, where the two sides measuring 6 feet and 8 feet are perpendicular to each other. What is the length of the diagonal side of the deck, which represents the hypotenuse?

1. 10 feet ✓
2. 12 feet
3. 14 feet
4. 16 feet

Why: To find the length of the hypotenuse (c) in a right triangle, we use the Pythagorean Theorem: a^2 + b^2 = c^2. Here, a = 6 feet and b = 8 feet. So, we calculate 6^2 + 8^2 = 36 + 64 = 100. Taking the square root of 100 gives us c = 10 feet. Therefore, the length of the diagonal side of the deck is 10 feet.

Emma is helping her father with a project in their yard in Houston, Texas. They need to place a ladder against the side of their two-story house, which is 12 feet high. If the ladder is 13 feet long, how far is the base of the ladder from the house? Use the Pythagorean Theorem to determine this distance.

1. 5 feet ✓
2. 7 feet
3. 8 feet
4. 10 feet

Why: To find the distance from the base of the ladder to the house, we can use the Pythagorean Theorem, which states that a^2 + b^2 = c^2, where c is the hypotenuse (the ladder), and a and b are the legs of the right triangle (the height of the house and the distance from the house). Here, the height of the house is 12 feet (a), and the length of the ladder is 13 feet (c). We need to find b (the distance from the house). So we have: a^2 + b^2 = c^2, which translates to 12^2 + b^2 = 13^2. That gives us 144 + b^2 = 169. Solving for b^2, we get b^2 = 169 - 144, which is b^2 = 25. Therefore, b = sqrt(25), which is 5 feet.

Mariana is helping her family at their cotton farm in Texas. They need to measure the distance from one corner of their rectangular field to the opposite corner. The field measures 24 meters in width and 10 meters in length. Using the Pythagorean Theorem, what is the distance between the two corners of the field?

1. 26 meters ✓
2. 28 meters
3. 30 meters
4. 34 meters

Why: To find the distance between the two corners of the rectangular field, we can use the Pythagorean Theorem, which states that in a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse. Here, a = 24 meters and b = 10 meters. Calculating: 24^2 + 10^2 = 576 + 100 = 676. Now, take the square root of 676 to find c: √676 = 26. Therefore, the distance from one corner to the opposite corner of the field is 26 meters.

Cristian is helping his grandfather build a wooden frame to support a climbing vine in their backyard in Austin, Texas. The frame will be in the shape of a right triangle. One leg of the triangle will be 6 feet long, and the other leg will be 8 feet long. What is the approximate length of the diagonal support beam needed to complete the frame?

1. 10 feet ✓
2. 14 feet
3. 12 feet
4. 11 feet

Why: To find the length of the diagonal support beam, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Here, a = 6 feet and b = 8 feet. So, we calculate c as follows: c = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 feet. Therefore, the approximate length of the diagonal support beam needed is 10 feet.

Mia is helping her uncle build a small fence around a rectangular garden in Austin, Texas. The length of the garden is 12 feet, and the width is 9 feet. To reinforce the garden’s corners, they plan to place a diagonal brace from one corner to the opposite corner. What is the length of the diagonal brace needed to complete the garden's support system?

1. 15 feet ✓
2. 21 feet
3. 108 feet
4. 27 feet

Why: To find the length of the diagonal brace, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (d) is equal to the sum of the squares of the lengths of the other two sides (a and b). Here, a = 12 feet and b = 9 feet. So, we calculate: d² = 12² + 9², which simplifies to d² = 144 + 81, giving us d² = 225. Taking the square root gives d = √225 = 15 feet. Therefore, the length of the diagonal brace needed is 15 feet.

Imani is helping her grandfather measure the distance between two points in a cotton field in Lubbock, Texas. One point is located at the corner of the field, and the other is 24 feet directly east and 10 feet north of that corner. What is the straight-line distance between the two points in feet?

1. 26 ✓
2. 28
3. 30
4. 32

Why: To find the straight-line distance between the two points, we can use the Pythagorean Theorem. The two points form a right triangle where one leg is 24 feet (east) and the other leg is 10 feet (north). According to the Pythagorean Theorem, we calculate the hypotenuse (distance) as follows: distance = sqrt(24^2 + 10^2). First, we compute 24^2 = 576 and 10^2 = 100. Then, we add these: 576 + 100 = 676. Finally, we take the square root of 676, which gives us 26. Therefore, the straight-line distance is 26 feet.

Michael is helping his father build a new bike ramp in their backyard in Brownsville, Texas. The ramp will form a right triangle where one side is 4 feet tall and the other side is 3 feet long. How long will the diagonal ramp be? Use the Pythagorean Theorem to find the length of the ramp in feet.

1. 5 ✓
2. 7
3. 6
4. 8

Why: To find the length of the diagonal ramp, we can use the Pythagorean Theorem, which states that a² + b² = c², where c is the hypotenuse. Here, a = 4 and b = 3. So, we have 4² + 3² = c², which simplifies to 16 + 9 = c². Therefore, 25 = c². Taking the square root of both sides gives us c = 5. The length of the ramp is 5 feet.