QuestionWhich statement about a $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$ triangle with hypotenuse $\sqrt{2}$ is true?

Studdy Solution

STEP 1

Assumptions1. We are dealing with a special right triangle with angles $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{}$ . This is also known as a45-45-90 triangle. . The length of the hypotenuse of this triangle is given as $\sqrt{}$ .
3. We are asked to verify which of the given statements is true.

STEP 2

The properties of a45-45-90 triangle are as follows1. The triangle is isosceles, meaning the two legs are of equal length.
2. The length of the hypotenuse is $\sqrt{2}$ times the length of each leg.
Let's denote the length of each leg as $a$ . Then, the length of the hypotenuse is $a\sqrt{2}$ .

STEP 3

Now, we can compare these properties with each of the given statements.
Statement A The length of the leg opposite either of the $\frac{\pi}{}$ angles of a special $\frac{\pi}{}, \frac{\pi}{}, \frac{\pi}{2}$ right triangle with a hypotenuse of $\sqrt{2}$ is equal to1.
We know that the length of the hypotenuse is $a\sqrt{2}$ . Given that the hypotenuse is $\sqrt{2}$ , we can set up the equation $a\sqrt{2} = \sqrt{2}$ .

STEP 4

olving the equation from step3 for $a$ gives us the length of each leg.
$a\sqrt{2} = \sqrt{2}$ $a = \frac{\sqrt{2}}{\sqrt{2}}$ $a =1$ So, statement A is true.

STEP 5

Statement B The length of a leg of a special $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$ right triangle is equal to half of the length of the hypotenuse.
We know that the length of the hypotenuse is $a\sqrt{2}$ , and we just found that $a =1$ . So, the length of the hypotenuse is $\sqrt{2}$ , and half of this is $\frac{\sqrt{2}}{2}$ , not1. So, statement B is false.

STEP 6

Statement C It is not possible for a special $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$ right triangle to be isosceles.
We know from the properties of a45-45-90 triangle that it is isosceles. So, statement C is false.

STEP 7

Statement D The length of the hypotenuse of a special $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$ right triangle is equal to twice the length of a leg opposite the $\frac{\pi}{4}$ angle.
We know that the length of the hypotenuse is $a\sqrt{2}$ , and we found that $a =1$ . So, the length of the hypotenuse is $\sqrt{2}$ , not2. So, statement D is false.
The only true statement is A The length of the leg opposite either of the $\frac{\pi}{4}$ angles of a special $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$ right triangle with a hypotenuse of $\sqrt{2}$ is equal to1.

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QuestionWhich statement about a $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$ triangle with hypotenuse $\sqrt{2}$ is true?

Studdy Solution

STEP 1

Assumptions1. We are dealing with a special right triangle with angles $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{}$ . This is also known as a45-45-90 triangle. . The length of the hypotenuse of this triangle is given as $\sqrt{}$ .
3. We are asked to verify which of the given statements is true.

STEP 2

The properties of a45-45-90 triangle are as follows1. The triangle is isosceles, meaning the two legs are of equal length.
2. The length of the hypotenuse is $\sqrt{2}$ times the length of each leg.
Let's denote the length of each leg as $a$ . Then, the length of the hypotenuse is $a\sqrt{2}$ .

STEP 3

STEP 4

olving the equation from step3 for $a$ gives us the length of each leg.
$a\sqrt{2} = \sqrt{2}$ $a = \frac{\sqrt{2}}{\sqrt{2}}$ $a =1$ So, statement A is true.

STEP 5

STEP 6

Statement C It is not possible for a special $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$ right triangle to be isosceles.
We know from the properties of a45-45-90 triangle that it is isosceles. So, statement C is false.

STEP 7

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QuestionWhich statement about a π4,π4,π2\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}4π​,4π​,2π​ triangle with hypotenuse 2\sqrt{2}2​ is true?

Studdy Solution

STEP 1

STEP 2

The properties of a45-45-90 triangle are as follows1. The triangle is isosceles, meaning the two legs are of equal length.2. The length of the hypotenuse is 2\sqrt{2}2​ times the length of each leg.Let's denote the length of each leg as aaa. Then, the length of the hypotenuse is a2a\sqrt{2}a2​.

STEP 3

STEP 4

olving the equation from step3 for aaa gives us the length of each leg.a2=2a\sqrt{2} = \sqrt{2}a2​=2​a=22a = \frac{\sqrt{2}}{\sqrt{2}}a=2​2​​a=1a =1a=1So, statement A is true.

STEP 5

STEP 6

Statement C It is not possible for a special π4,π4,π2\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}4π​,4π​,2π​ right triangle to be isosceles.We know from the properties of a45-45-90 triangle that it is isosceles. So, statement C is false.

STEP 7

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QuestionWhich statement about a $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$ triangle with hypotenuse $\sqrt{2}$ is true?

The properties of a45-45-90 triangle are as follows1. The triangle is isosceles, meaning the two legs are of equal length.
2. The length of the hypotenuse is $\sqrt{2}$ times the length of each leg.
Let's denote the length of each leg as $a$ . Then, the length of the hypotenuse is $a\sqrt{2}$ .

olving the equation from step3 for $a$ gives us the length of each leg.
$a\sqrt{2} = \sqrt{2}$ $a = \frac{\sqrt{2}}{\sqrt{2}}$ $a =1$ So, statement A is true.

Statement C It is not possible for a special $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$ right triangle to be isosceles.
We know from the properties of a45-45-90 triangle that it is isosceles. So, statement C is false.