Chapter 6 Triangles

Case Study 1: Land Division for a Park

Case Description:
A city council has allocated a triangular plot of land for the construction of a public park. The park will have three distinct sections: a children’s play area, a garden, and a pond. The plot is in the shape of a right-angled triangle with the right angle at $B$, where $AB = 24 \, \text{m}$, $BC = 18 \, \text{m}$, and $AC$ represents the hypotenuse. The council wants to partition the plot into smaller triangles and allocate sections proportionately. To estimate materials, the council needs to calculate the length of the hypotenuse, the area of the plot, and whether similar triangles can be constructed within the plot.

MCQs:

1. What is the length of the hypotenuse $AC$ of the plot?

   • A) 26 m
   • B) 30 m
   • C) 32 m
   • D) 28 m

2. What is the area of the triangular plot?

   • A) 216 m²
   • B) 234 m²
   • C) 252 m²
   • D) 270 m²

3. If the council divides the plot into two smaller triangles by drawing a line parallel to $AB$ through point $D$ on $AC$, which similarity criterion applies to the two triangles created?

   • A) AA Similarity
   • B) SAS Similarity
   • C) RHS Similarity
   • D) SSS Similarity

4. If the length $AB$ were doubled, what would the new hypotenuse $AC'$ be for the enlarged triangle?

   • A) 30 m
   • B) 36 m
   • C) 42 m
   • D) 52 m

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Case Study 2: Bridge Construction Over a River

Case Description:
A construction company is designing a bridge over a river. The bridge will have supports on both banks and cables connecting the top of the supports to points along the bridge to ensure stability. The bridge forms a right triangle, where one leg represents the height of the support, the other leg represents the horizontal span across the river, and the hypotenuse represents the supporting cable. If the height of the support is 40 m and the span across the river is 30 m, the engineers need to calculate the length of the cable, the angle at the support with the riverbank, and the possible similarity between this right triangle and other parts of the bridge structure for balance.

MCQs:

1. What is the length of the hypotenuse (the supporting cable)?

   • A) 45 m
   • B) 50 m
   • C) 55 m
   • D) 60 m

2. What is the measure of the angle between the support and the cable to the nearest degree?

   • A) 37°
   • B) 53°
   • C) 48°
   • D) 60°

3. Which similarity criterion is used if another right triangle is constructed with a height of 20 m and a span of 15 m?

   • A) AA Similarity
   • B) RHS Similarity
   • C) SSS Similarity
   • D) SAS Similarity

4. What would the hypotenuse of this new smaller triangle be?

   • A) 20 m
   • B) 25 m
   • C) 30 m
   • D) 35 m

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Case Study 3: Triangular Flower Beds in a Garden

Case Description:
A landscaper is designing triangular flower beds in a garden. There are three flower beds, each of a different size but similar in shape to the largest one. The largest flower bed has side lengths of 10 m, 12 m, and 14 m. The landscaper needs to calculate the areas of the smaller flower beds if they are similar by a scale factor of 1/2 and 1/3 of the largest flower bed. The landscaper also wants to ensure the angles remain the same across all three flower beds.

MCQs:

1. What is the perimeter of the largest flower bed?

   • A) 34 m
   • B) 36 m
   • C) 38 m
   • D) 40 m

2. What would be the area of the flower bed scaled down by a factor of 1/2?

   • A) 24 m²
   • B) 28 m²
   • C) 30 m²
   • D) 21 m²

3. If the largest flower bed has angles 45°, 60°, and 75°, what would be the angles of the smallest flower bed (1/3 scale factor)?

   • A) 15°, 20°, and 25°
   • B) 45°, 60°, and 75°
   • C) 30°, 40°, and 50°
   • D) 35°, 55°, and 65°

4. What is the area of the flower bed scaled down by a factor of 1/3?

   • A) 14 m²
   • B) 12 m²
   • C) 10 m²
   • D) 18 m²

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Case Study 4: Mountain Hiking Paths

Case Description:
An adventure company is mapping out hiking paths on a mountain. They plan three routes, each forming a right triangle with the mountain slope. Route A has a height of 5 km and a base of 12 km. The team is interested in estimating the length of the hiking trail along the hypotenuse, determining the inclination angle at the base, and planning two similar paths, Route B and Route C, with the same angles but reduced in size by factors of 1/2 and 1/4, respectively.

MCQs:

1. What is the length of the hypotenuse for Route A?

   • A) 12 km
   • B) 13 km
   • C) 15 km
   • D) 17 km

2. What is the measure of the angle at the base of Route A?

   • A) 30°
   • B) 45°
   • C) 22°
   • D) 67°

3. What is the hypotenuse of Route B if it is reduced by a factor of 1/2?

   • A) 6 km
   • B) 7.5 km
   • C) 9 km
   • D) 8 km

4. What will be the hypotenuse of Route C if it is reduced by a factor of 1/4?

   • A) 3.75 km
   • B) 4 km
   • C) 5 km
   • D) 5.5 km

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Case Study 5: Construction of a Triangular Billboard

Case Description:
A company is constructing a large triangular billboard that needs three supporting beams. The billboard is designed to be an equilateral triangle with each side measuring 15 meters. The company plans to install similar but smaller versions of this billboard in two other locations, with side lengths reduced by half and one-third. The company needs to calculate the perimeter, area, and other relevant details for each of these smaller billboards to ensure uniformity in construction.

MCQs:

1. What is the perimeter of the largest billboard?

   • A) 30 m
   • B) 45 m
   • C) 50 m
   • D) 60 m

2. What is the area of the largest billboard?

   • A) 93.5 m²
   • B) 97.4 m²
   • C) 98.5 m²
   • D) 95.2 m²

3. What will be the side length of the billboard reduced by a factor of 1/2?

   • A) 7.5 m
   • B) 10 m
   • C) 8 m
   • D) 9 m

4. What will be the area of the billboard reduced by a factor of 1/3?

   • A) 15 m²
   • B) 18 m²
   • C) 20 m²
   • D) 16.3 m²