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The longest side of an acute isosceles triangle is 12 centimeters. Rounded to the nearest tenth, what is the smallest possible length of one of the two congruent sides? 6.0 cm O 6.1 cm O8.4 cm 8.5 cm

The longest side of an acute isosceles triangle is 12 centimeters. Rounded to the nearest tenth, what is the smallest possible length of one of the two congruent sides? 6.0 cm O 6.1 cm O8.4 cm 8.5 cm

Mathematics For Machine Technology
Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter62: Volumes Of Prisms And Cylinders
Section: Chapter Questions
Problem 38A: A copper casting is in the shape of a prism with an equilateral triangle base. The length of each...
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**Educational Content** ### Question: The longest side of an acute isosceles triangle is 12 centimeters. Rounded to the nearest tenth, what is the smallest possible length of one of the two congruent sides? - 6.0 cm - 6.1 cm - 8.4 cm - 8.5 cm ### Explanation: To solve this problem, it’s important to understand the properties of an acute isosceles triangle. An isosceles triangle has two sides of equal length. In an acute triangle, all angles are less than 90 degrees. Given that the longest side (the base) is 12 centimeters, we need to determine if the possible lengths for the two congruent sides maintain the acuteness. By using the triangle inequality principle (sum of any two sides of a triangle must be greater than the third side), we can evaluate the possible lengths: 1. Let's denote the length of each of the congruent sides as \( x \). 2. According to the triangle inequality theorem for an acute triangle: - The sum of any two sides must be greater than the third side. - \( x + x > 12 \) (sum of the two congruent sides greater than the longest side). - \( x + 12 > x \) and \( x + 12 > x \) (each congruent side with the longest side greater than the remaining side). The sides also need to keep the triangle acute: 3. For an isosceles triangle to be acute, the square of the longest side must be less than the sum of the squares of the two equal sides: \( 12^2 < 2x^2 \), which simplifies to: \( 144 < 2x^2 \). By solving this inequality: \( 72 < x^2 \). Therefore, \( x > \sqrt{72} \). \( x > 8.485 \) (approximately). By comparing this result with the given options, rounded to the nearest tenth, the answer is: - 8.5 cm (smallest possible length for one of the congruent sides to maintain an acute triangle). ### Multiple Choices and Correct Answer: - 6.0 cm - 6.1 cm - 8.4 cm - **8.5 cm** (correct answer) The correct answer is
Transcribed Image Text:**Educational Content** ### Question: The longest side of an acute isosceles triangle is 12 centimeters. Rounded to the nearest tenth, what is the smallest possible length of one of the two congruent sides? - 6.0 cm - 6.1 cm - 8.4 cm - 8.5 cm ### Explanation: To solve this problem, it’s important to understand the properties of an acute isosceles triangle. An isosceles triangle has two sides of equal length. In an acute triangle, all angles are less than 90 degrees. Given that the longest side (the base) is 12 centimeters, we need to determine if the possible lengths for the two congruent sides maintain the acuteness. By using the triangle inequality principle (sum of any two sides of a triangle must be greater than the third side), we can evaluate the possible lengths: 1. Let's denote the length of each of the congruent sides as \( x \). 2. According to the triangle inequality theorem for an acute triangle: - The sum of any two sides must be greater than the third side. - \( x + x > 12 \) (sum of the two congruent sides greater than the longest side). - \( x + 12 > x \) and \( x + 12 > x \) (each congruent side with the longest side greater than the remaining side). The sides also need to keep the triangle acute: 3. For an isosceles triangle to be acute, the square of the longest side must be less than the sum of the squares of the two equal sides: \( 12^2 < 2x^2 \), which simplifies to: \( 144 < 2x^2 \). By solving this inequality: \( 72 < x^2 \). Therefore, \( x > \sqrt{72} \). \( x > 8.485 \) (approximately). By comparing this result with the given options, rounded to the nearest tenth, the answer is: - 8.5 cm (smallest possible length for one of the congruent sides to maintain an acute triangle). ### Multiple Choices and Correct Answer: - 6.0 cm - 6.1 cm - 8.4 cm - **8.5 cm** (correct answer) The correct answer is
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