4. the lengths of two sides of a triangle are given. determine the two lengths the third side be between.
a. 18 yd, 16 yd
b. 6.5 meters, 6.5 meters
5. ivan knows two sides of a triangle are 16 meters and 6 meters. what two values must the third side be between?
6. carla and marty each have three straws. the straws have lengths 8 inches, 10 inches and 14 inches. they are going to use their straws to form a triangle. carla predicts that they will be able to make a non - right triangle. marty predicts that the triangle will be a right triangle. who is correct? explain your reasoning.

Question

4. the lengths of two sides of a triangle are given. determine the two lengths the third side be between.
   a. 18 yd, 16 yd
   b. 6.5 meters, 6.5 meters
5. ivan knows two sides of a triangle are 16 meters and 6 meters. what two values must the third side be between?
6. carla and marty each have three straws. the straws have lengths 8 inches, 10 inches and 14 inches. they are going to use their straws to form a triangle. carla predicts that they will be able to make a non - right triangle. marty predicts that the triangle will be a right triangle. who is correct? explain your reasoning.

Accepted Answer

### Problem 4a: # Explanation: ## Step1: Recall triangle inequality theorem For a triangle with sides $a$, $b$, $c$, the third side $c$ must satisfy $|a - b| < c < a + b$. Here, $a = 18$ yd, $b = 16$ yd. ## Step2: Calculate the range First, find the difference: $|18 - 16| = 2$ yd. Then, find the sum: $18 + 16 = 34$ yd. So the third side $c$ satisfies $2$ yd $< c < 34$ yd. # Answer: The third side is between $2$ yd and $34$ yd. ### Problem 4b: # Explanation: ## Step1: Apply triangle inequality theorem Let $a = 65$ m, $b = 65$ m. The third side $c$ must satisfy $|a - b| < c < a + b$. ## Step2: Compute the range Difference: $|65 - 65| = 0$ m. Sum: $65 + 65 = 130$ m. So $0$ m $< c < 130$ m (but since side length is positive, effectively $0 < c < 130$, and in context of triangle, $c>0$ and $c < 130$, also $c$ must be positive, so the third side is between $0$ m (exclusive) and $130$ m (exclusive), but more precisely, since sides are positive, $0 < c < 130$, and for a valid triangle, $c$ must be greater than $0$ and less than $130$. However, since the two sides are equal (isosceles triangle), the third side must be greater than $0$ and less than $130$ meters. # Answer: The third side is between $0$ meters (exclusive) and $130$ meters (exclusive), or more precisely, between $0$ m and $130$ m (with $c>0$ and $c < 130$). ### Problem 5: # Explanation: ## Step1: Use triangle inequality theorem Given $a = 16$ m, $b = 6$ m. The third side $c$ satisfies $|a - b| < c < a + b$. ## Step2: Calculate the bounds Difference: $|16 - 6| = 10$ m. Sum: $16 + 6 = 22$ m. So $10$ m $< c < 22$ m. # Answer: The third side is between $10$ meters and $22$ meters. ### Problem 6: # Explanation: ## Step1: Recall Pythagorean theorem For a right triangle, $a^2 + b^2 = c^2$ (where $c$ is the hypotenuse, the longest side). The sides are $8$ in, $10$ in, $14$ in. Let's check if $8^2 + 10^2 = 14^2$. ## Step2: Compute the squares $8^2 = 64$, $10^2 = 100$, sum: $64 + 100 = 164$. $14^2 = 196$. Since $164 eq196$, it's not a right triangle. So Carla is correct. # Answer: Carla is correct because $8^2 + 10^2 = 64 + 100 = 164 eq14^2 = 196$, so the triangle is not a right triangle.

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QUESTION IMAGE

Question

Explanation:

Problem 4a:

Step1: Recall triangle inequality theorem

Step2: Calculate the range

Step1: Apply triangle inequality theorem

Step2: Compute the range

Step1: Use triangle inequality theorem

Step2: Calculate the bounds

Answer:

Problem 4b: