TEKS 8.4A Worksheets — Grade 8 Use similar right triangles to develop an

Liam is planning a road trip from Austin to San Antonio, which are 80 miles apart. He wants to stop at a restaurant that is 1/4 of the way to his destination. If he drives at a speed of 60 miles per hour, how long will it take him to reach the restaurant?

1. 20 minutes ✓
2. 15 minutes
3. 30 minutes
4. 25 minutes

Why: To find out how far Liam needs to drive to reach the restaurant, calculate 1/4 of the total distance: 80 miles * 1/4 = 20 miles. Next, to find out how long it will take him to drive 20 miles at a speed of 60 miles per hour, use the formula time = distance / speed: time = 20 miles / 60 miles per hour = 1/3 hour or 20 minutes.

Gabriela is studying the growth of Texas wildflowers in her garden. She measured the height of her sunflowers at two different times. At the first measurement, the height was 120 centimeters, and after 25 days, the height increased to 225 centimeters. What is the slope, or rate of growth per day, of Gabriela's sunflowers based on these measurements?

1. 4.2 ✓
2. 5
3. 3.5
4. 6

Why: To find the slope, we use the formula m = (y2 - y1) / (x2 - x1). Here, y1 = 120 cm, y2 = 225 cm, x1 = 0 days, and x2 = 25 days. Therefore, m = (225 - 120) / (25 - 0) = 105 / 25 = 4.2 cm per day. Thus, the correct answer is 4.2.

Lucas is helping his family run a peach orchard in the Rio Grande Valley. They found that for every 3 peach trees planted, they can expect to harvest approximately 45 pounds of peaches. If Lucas's family plants 12 trees this season, how many pounds of peaches can they expect to harvest? Use the ratio from the trees to the expected harvest to solve the problem.

1. 180 ✓
2. 135
3. 60
4. 90

Why: To find the expected harvest for 12 trees, we first determine the ratio of trees to pounds of peaches. Since 3 trees yield 45 pounds, we can set up a proportion: 3/45 = 12/x. Cross-multiplying gives us 3x = 540. Dividing both sides by 3 results in x = 180. Therefore, if they plant 12 trees, they can expect to harvest 180 pounds of peaches.

Skylar is creating a poster for a school project on Texas agriculture, and she is using a map of Texas to show the distribution of peach farms. She marks two peach farms on the map: the first farm is located at the coordinates (2, 30) and the second farm is located at (6, 50). What is the slope of the line connecting these two farms, which represents the rate of change in the number of peaches produced per mile east on the map?

1. 5 ✓
2. 10
3. 4
4. 3

Why: To find the slope (m), use the formula (y2 - y1) / (x2 - x1). Here, the coordinates are (x1, y1) = (2, 30) and (x2, y2) = (6, 50). Substitute the values: m = (50 - 30) / (6 - 2) = 20 / 4 = 5. Thus, the slope is 5, which means for every mile east, the production of peaches increases by 5 units.

Luis is studying the slope of a hill in the Texas Hill Country where he enjoys hiking. He measures two points on the slope: Point A is located at (2, 4) and Point B is located at (6, 10). Which expression best represents the slope of the hill between these two points?

1. 3/2 ✓
2. 1/2
3. 2
4. 1

Why: To find the slope (m) between two points (x1, y1) and (x2, y2), use the formula m = (y2 - y1) / (x2 - x1). For points A (2, 4) and B (6, 10), substitute the values: m = (10 - 4) / (6 - 2) = 6 / 4 = 3/2. Thus, the correct answer is 3/2.

In Austin, Tyrese is designing a mural that represents the Texas Hill Country. He plans to use similar right triangles to determine the height of the mural. If one triangle has a height of 6 feet and a base of 4 feet, and another triangle's base is 10 feet, what will be the height of the second triangle? Which equation can be used to find the height of the second triangle (h)?

1. h = 10 * (6/4) ✓
2. h = 10 / (6/4)
3. h = 4 * (10/6)
4. h = 4 / (10/6)

Why: To find the height of the second triangle, we can setup a proportion based on the similar triangles. The ratio of height to base for the first triangle is 6/4. Therefore, we can represent the height (h) of the second triangle as h = base * (height/base of first triangle). So, h = 10 * (6/4). This calculates to h = 10 * 1.5 = 15 feet. Hence, the equation h = 10 * (6/4) correctly determines the height.

Camille is designing a mini golf course in Austin, Texas, and wants to create a hole that has an uphill slope. She marked two points on the hill. The first point is at (2, 3) and the second point is at (6, 7). What is the slope of the line connecting these two points?

1. 1 ✓
2. 1/2
3. 2
4. 4/3

Why: To find the slope (m) between the two points (x1, y1) = (2, 3) and (x2, y2) = (6, 7), we use the formula m = (y2 - y1) / (x2 - x1). First, we calculate the change in y-values: 7 - 3 = 4. Next, we calculate the change in x-values: 6 - 2 = 4. Now, we divide the change in y by the change in x: m = 4 / 4 = 1. Thus, the slope of the line is 1.

Luis is building a fence around his garden in McAllen, Texas. He measures two points along the fence. The first point is at (2, 4) feet, and the second point is at (6, 10) feet. What is the slope of the line connecting these two points, which represents the rate of change in height compared to the change in length?

1. 3/2 ✓
2. 1/2
3. 2
4. 1

Why: To find the slope (m) of the line connecting the two points (x1, y1) = (2, 4) and (x2, y2) = (6, 10), we use the slope formula: m = (y2 - y1) / (x2 - x1). Plugging in the values, we get m = (10 - 4) / (6 - 2) = 6 / 4 = 3 / 2. Therefore, the slope is 3/2.