University:
California State University, Northridge
Course:
MATH 262 | Introduction to Linear Algebra
Academic year:

2024
Views:

318

Pages:

3
Author:

Beckett Gray

Question 1 Point B is a point of tangency. Find the radius 𝑟 r of 𝑂 𝐶 OC. Answer Since point B is a point of tangency, 𝐴 𝐵 ⊥ 𝐵 𝐶 AB⊥BC, making △ 𝐴 𝐵 𝐶 △ABC a right triangle. Using the Pythagorean Theorem: 𝐴 𝐵 2 + 𝐵 𝐶 2 = 𝐴 𝐶 2 AB 2 +BC 2 =AC 2 Given: 𝐴 𝐵 = 14 AB=14, 𝐵 𝐶 = 𝑟 BC=r, 𝐴 𝐶 = 𝑟 + 7 AC=r+7. Substitute into the equation: 1 4 2 + 𝑟 2 = ( 𝑟 + 7 ) 2 . 14 2 +r 2 =(r+7) 2 . Simplify: 196 + 𝑟 2 = 𝑟 2 + 14 𝑟 + 49. 196+r 2 =r 2 +14r+49. 196 − 49 = 14 𝑟 . 196−49=14r. 147 = 14 𝑟 . 147=14r. 𝑟 = 10.5. r=10.5. Final Answer: 𝑟 = 10.5 r=10.5. Question 2 Determine the area, in square feet, of the smallest square that can contain a circle with a radius of 8 feet. Answer The side of the square is equal to the diameter of the circle: Diameter = 2 × radius = 2 × 8 = 16 feet . Diameter=2×radius=2×8=16feet. The area of the square is: 𝐴 = 𝑠 2 = 1 6 2 = 256 square feet . A=s 2 =16 2 =256square feet. Final Answer: 256 square feet 256square feet. Question 3 Are all circles similar? Explain your reasoning. Answer Step 1: To explain similarity. Two shapes are similar if they have the same shape but can differ in size. Step 2: All circles have the same shape regardless of their size. Since they are geometrically identical except for scaling, all circles are similar. Final Answer: Yes, all circles are similar. Question 4 For the circles described and shown, how many common tangents do they possess? (Internally tangent circles) Answer Step 1: Analyze the number of tangents. For two internally tangent circles, there is exactly 1 common tangent. Step 2: The diagram confirms the existence of one tangent line touching both circles. Final Answer: 1 common tangent. Question 5 Are three circles with radii of 20 units (inner) and 23 units (outer) similar? Why or why not? Answer Step 1: To determine whether the circles are similar. Step 2: Explanation: Similarity is determined by shape, not size. All circles have the same shape, regardless of their radii. Thus, any circle can be scaled to match another circle. Final Answer: Yes, the given circles are similar.