Math / Geometry

QuestionSolve for the long leg and for the hypotenuse of the $30-60-90$ thangle. a. $a=20$ and $b=10$ b. $a=20$ and $b=10 \sqrt{3}$ c. $a=10$ and $b=10 \sqrt{3}$ d. $a=10 \sqrt{3}$ and $b=20$

Studdy Solution

STEP 1

1. We are dealing with a $30^\circ-60^\circ-90^\circ$ triangle, which is a special right triangle.
2. The sides of a $30^\circ-60^\circ-90^\circ$ triangle have a known ratio: the short leg (opposite the $30^\circ$ angle) is $x$ , the long leg (opposite the $60^\circ$ angle) is $x\sqrt{3}$ , and the hypotenuse is $2x$ .
3. We are given different values for $a$ and $b$ in each part, which represent the lengths of the legs.

STEP 2

1. Identify the given sides and their roles in the $30^\circ-60^\circ-90^\circ$ triangle.
2. Use the known ratios of the sides to solve for the unknown sides.
3. Verify the results for each case.

STEP 3

For each part, identify which side $a$ and $b$ represent in the $30^\circ-60^\circ-90^\circ$ triangle. Recall that the short leg is $x$ , the long leg is $x\sqrt{3}$ , and the hypotenuse is $2x$ .

STEP 4

For part (a), $a = 20$ and $b = 10$ . We need to determine which side is the short leg and which is the long leg or hypotenuse.

STEP 5

In part (a), if $a = 20$ is the short leg, then the long leg should be $20\sqrt{3}$ , and the hypotenuse should be $40$ . Since $b = 10$ , this does not match the expected ratio, so $a = 20$ must be the hypotenuse. Thus, the short leg is $10$ and the long leg is $10\sqrt{3}$ .

STEP 6

For part (b), $a = 20$ and $b = 10\sqrt{3}$ . Determine which sides these represent.

STEP 7

In part (b), if $a = 20$ is the hypotenuse, then the short leg should be $10$ and the long leg should be $10\sqrt{3}$ . Since $b = 10\sqrt{3}$ , this matches the expected ratio. Therefore, the short leg is $10$ , the long leg is $10\sqrt{3}$ , and the hypotenuse is $20$ .

STEP 8

For part (c), $a = 10$ and $b = 10\sqrt{3}$ . Identify the sides.

STEP 9

In part (c), if $a = 10$ is the short leg, then the long leg should be $10\sqrt{3}$ and the hypotenuse should be $20$ . Since $b = 10\sqrt{3}$ , this matches the expected ratio. Therefore, the short leg is $10$ , the long leg is $10\sqrt{3}$ , and the hypotenuse is $20$ .

STEP 10

For part (d), $a = 10\sqrt{3}$ and $b = 20$ . Determine which sides these represent.

STEP 11

In part (d), if $a = 10\sqrt{3}$ is the long leg, then the short leg should be $10$ and the hypotenuse should be $20$ . Since $b = 20$ , this matches the expected ratio. Therefore, the short leg is $10$ , the long leg is $10\sqrt{3}$ , and the hypotenuse is $20$ .
The solutions for each part are: a. Short leg: $10$ , Long leg: $10\sqrt{3}$ , Hypotenuse: $20$ b. Short leg: $10$ , Long leg: $10\sqrt{3}$ , Hypotenuse: $20$ c. Short leg: $10$ , Long leg: $10\sqrt{3}$ , Hypotenuse: $20$ d. Short leg: $10$ , Long leg: $10\sqrt{3}$ , Hypotenuse: $20$

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QuestionSolve for the long leg and for the hypotenuse of the $30-60-90$ thangle. a. $a=20$ and $b=10$ b. $a=20$ and $b=10 \sqrt{3}$ c. $a=10$ and $b=10 \sqrt{3}$ d. $a=10 \sqrt{3}$ and $b=20$

Studdy Solution

STEP 1

STEP 2

1. Identify the given sides and their roles in the $30^\circ-60^\circ-90^\circ$ triangle.
2. Use the known ratios of the sides to solve for the unknown sides.
3. Verify the results for each case.

STEP 3

For each part, identify which side $a$ and $b$ represent in the $30^\circ-60^\circ-90^\circ$ triangle. Recall that the short leg is $x$ , the long leg is $x\sqrt{3}$ , and the hypotenuse is $2x$ .

STEP 4

For part (a), $a = 20$ and $b = 10$ . We need to determine which side is the short leg and which is the long leg or hypotenuse.

STEP 5

In part (a), if $a = 20$ is the short leg, then the long leg should be $20\sqrt{3}$ , and the hypotenuse should be $40$ . Since $b = 10$ , this does not match the expected ratio, so $a = 20$ must be the hypotenuse. Thus, the short leg is $10$ and the long leg is $10\sqrt{3}$ .

STEP 6

For part (b), $a = 20$ and $b = 10\sqrt{3}$ . Determine which sides these represent.

STEP 7

In part (b), if $a = 20$ is the hypotenuse, then the short leg should be $10$ and the long leg should be $10\sqrt{3}$ . Since $b = 10\sqrt{3}$ , this matches the expected ratio. Therefore, the short leg is $10$ , the long leg is $10\sqrt{3}$ , and the hypotenuse is $20$ .

STEP 8

For part (c), $a = 10$ and $b = 10\sqrt{3}$ . Identify the sides.

STEP 9

In part (c), if $a = 10$ is the short leg, then the long leg should be $10\sqrt{3}$ and the hypotenuse should be $20$ . Since $b = 10\sqrt{3}$ , this matches the expected ratio. Therefore, the short leg is $10$ , the long leg is $10\sqrt{3}$ , and the hypotenuse is $20$ .

STEP 10

For part (d), $a = 10\sqrt{3}$ and $b = 20$ . Determine which sides these represent.

STEP 11

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QuestionSolve for the long leg and for the hypotenuse of the 30−60−9030-60-9030−60−90 thangle. a. a=20a=20a=20 and b=10b=10b=10 b. a=20a=20a=20 and b=103b=10 \sqrt{3}b=103​ c. a=10a=10a=10 and b=103b=10 \sqrt{3}b=103​ d. a=103a=10 \sqrt{3}a=103​ and b=20b=20b=20

Studdy Solution

STEP 1

STEP 2

1. Identify the given sides and their roles in the 30∘−60∘−90∘30^\circ-60^\circ-90^\circ30∘−60∘−90∘ triangle.2. Use the known ratios of the sides to solve for the unknown sides.3. Verify the results for each case.

STEP 3

For each part, identify which side aaa and bbb represent in the 30∘−60∘−90∘30^\circ-60^\circ-90^\circ30∘−60∘−90∘ triangle. Recall that the short leg is xxx, the long leg is x3x\sqrt{3}x3​, and the hypotenuse is 2x2x2x.

STEP 4

For part (a), a=20a = 20a=20 and b=10b = 10b=10. We need to determine which side is the short leg and which is the long leg or hypotenuse.

STEP 5

STEP 6

For part (b), a=20a = 20a=20 and b=103b = 10\sqrt{3}b=103​. Determine which sides these represent.

STEP 7

STEP 8

For part (c), a=10a = 10a=10 and b=103b = 10\sqrt{3}b=103​. Identify the sides.

STEP 9

STEP 10

For part (d), a=103a = 10\sqrt{3}a=103​ and b=20b = 20b=20. Determine which sides these represent.

STEP 11

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QuestionSolve for the long leg and for the hypotenuse of the $30-60-90$ thangle. a. $a=20$ and $b=10$ b. $a=20$ and $b=10 \sqrt{3}$ c. $a=10$ and $b=10 \sqrt{3}$ d. $a=10 \sqrt{3}$ and $b=20$

1. Identify the given sides and their roles in the $30^\circ-60^\circ-90^\circ$ triangle.
2. Use the known ratios of the sides to solve for the unknown sides.
3. Verify the results for each case.

For each part, identify which side $a$ and $b$ represent in the $30^\circ-60^\circ-90^\circ$ triangle. Recall that the short leg is $x$ , the long leg is $x\sqrt{3}$ , and the hypotenuse is $2x$ .

For part (a), $a = 20$ and $b = 10$ . We need to determine which side is the short leg and which is the long leg or hypotenuse.

For part (b), $a = 20$ and $b = 10\sqrt{3}$ . Determine which sides these represent.

For part (c), $a = 10$ and $b = 10\sqrt{3}$ . Identify the sides.

For part (d), $a = 10\sqrt{3}$ and $b = 20$ . Determine which sides these represent.