Could 10.5cm,8.0cm, and 4.0cm be the side lengths of a triangle? Choose 1 answer: (A) Yes (B) No

Check Triangle Inequality Theorem: To determine if three lengths can form a triangle, we can use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Calculate Sum of Two Shortest Sides: First, we check if the sum of the two shortest sides is greater than the longest side. The two shortest sides are 8.0cm and 4.0cm, and the longest side is 10.5cm. We calculate 8.0cm + 4.0cm and compare it to 10.5cm. Verify First Combination: The sum of the two shortest sides is 8.0cm + 4.0cm = 12.0cm, which is greater than the longest side, 10.5cm. This satisfies the Triangle Inequality Theorem for these two sides. Verify Second Combination: Next, we check the other two combinations to ensure they also satisfy the Triangle Inequality Theorem. We check if 10.5cm + 4.0cm is greater than 8.0cm and if 10.5cm + 8.0cm is greater than 4.0cm. Confirm Triangle Formation: The sum of 10.5cm and 4.0cm is 14.5cm, which is greater than 8.0cm. The sum of 10.5cm and 8.0cm is 18.5cm, which is greater than 4.0cm. Both of these sums satisfy the Triangle Inequality Theorem. Since all three combinations of side lengths satisfy the Triangle Inequality Theorem, the lengths 10.5cm, 8.0cm, and 4.0cm can indeed form a triangle.

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Is (x, y) a solution to the system of equations?

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Q. Could

10.5 \mathrm{~cm}, 8.0 \mathrm{~cm}

, and

4.0 \mathrm{~cm}

be the side lengths of a triangle?

\newline

Choose

1

answer:

\newline

(A) Yes

\newline

(B) No

Check Triangle Inequality Theorem: To determine if three lengths can form a triangle, we can use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Calculate Sum of Two Shortest Sides: First, we check if the sum of the two shortest sides is greater than the longest side. The two shortest sides are $8.0\,\text{cm}$ and $4.0\,\text{cm}$ , and the longest side is $10.5\,\text{cm}$ . We calculate $8.0\,\text{cm} + 4.0\,\text{cm}$ and compare it to $10.5\,\text{cm}$ .
Verify First Combination: The sum of the two shortest sides is $8.0\,\text{cm} + 4.0\,\text{cm} = 12.0\,\text{cm}$ , which is greater than the longest side, $10.5\,\text{cm}$ . This satisfies the Triangle Inequality Theorem for these two sides.
Verify Second Combination: Next, we check the other two combinations to ensure they also satisfy the Triangle Inequality Theorem. We check if $10.5\,\text{cm} + 4.0\,\text{cm}$ is greater than $8.0\,\text{cm}$ and if $10.5\,\text{cm} + 8.0\,\text{cm}$ is greater than $4.0\,\text{cm}$ .
Confirm Triangle Formation: The sum of $10.5\,\text{cm}$ and $4.0\,\text{cm}$ is $14.5\,\text{cm}$ , which is greater than $8.0\,\text{cm}$ . The sum of $10.5\,\text{cm}$ and $8.0\,\text{cm}$ is $18.5\,\text{cm}$ , which is greater than $4.0\,\text{cm}$ . Both of these sums satisfy the Triangle Inequality Theorem.
Confirm Triangle Formation: The sum of $10.5\,\text{cm}$ and $4.0\,\text{cm}$ is $14.5\,\text{cm}$ , which is greater than $8.0\,\text{cm}$ . The sum of $10.5\,\text{cm}$ and $8.0\,\text{cm}$ is $18.5\,\text{cm}$ , which is greater than $4.0\,\text{cm}$ . Both of these sums satisfy the Triangle Inequality Theorem. Since all three combinations of side lengths satisfy the Triangle Inequality Theorem, the lengths $10.5\,\text{cm}$ , $8.0\,\text{cm}$ , and $4.0\,\text{cm}$ can indeed form a triangle.