Math / Geometry

QuestionThis is a 45-45-90 triangle. What is the measure of $x$ ? $x=\sqrt{[?]}$

Studdy Solution

STEP 1

1. The triangle is a 45-45-90 triangle, meaning it is an isosceles right triangle.
2. In a 45-45-90 triangle, the legs are equal in length, and the hypotenuse is $x\sqrt{2}$ , where $x$ is the length of each leg.

STEP 2

1. Understand the properties of a 45-45-90 triangle.
2. Set up an equation based on the relationship between the legs and the hypotenuse.
3. Solve for $x$ .

STEP 3

Recognize that in a 45-45-90 triangle, the legs are equal, and the hypotenuse is $x\sqrt{2}$ , where $x$ is the length of each leg.

STEP 4

Set up the equation using the relationship of the hypotenuse to the legs in a 45-45-90 triangle. Given that the hypotenuse is $\sqrt{6}$ , we have:
$x\sqrt{2} = \sqrt{6}$

STEP 5

Solve for $x$ by dividing both sides of the equation by $\sqrt{2}$ :
$x = \frac{\sqrt{6}}{\sqrt{2}}$

STEP 6

Simplify the expression by rationalizing the denominator:
$x = \frac{\sqrt{6}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$ $= \frac{\sqrt{12}}{2}$

STEP 7

Simplify $\sqrt{12}$ to find the final expression for $x$ :
$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$
Thus,
$x = \frac{2\sqrt{3}}{2} = \sqrt{3}$
The measure of $x$ is:
$x = \sqrt{3}$

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QuestionThis is a 45-45-90 triangle. What is the measure of $x$ ? $x=\sqrt{[?]}$

Studdy Solution

STEP 1

1. The triangle is a 45-45-90 triangle, meaning it is an isosceles right triangle.
2. In a 45-45-90 triangle, the legs are equal in length, and the hypotenuse is $x\sqrt{2}$ , where $x$ is the length of each leg.

STEP 2

1. Understand the properties of a 45-45-90 triangle.
2. Set up an equation based on the relationship between the legs and the hypotenuse.
3. Solve for $x$ .

STEP 3

Recognize that in a 45-45-90 triangle, the legs are equal, and the hypotenuse is $x\sqrt{2}$ , where $x$ is the length of each leg.

STEP 4

Set up the equation using the relationship of the hypotenuse to the legs in a 45-45-90 triangle. Given that the hypotenuse is $\sqrt{6}$ , we have:
$x\sqrt{2} = \sqrt{6}$

STEP 5

Solve for $x$ by dividing both sides of the equation by $\sqrt{2}$ :
$x = \frac{\sqrt{6}}{\sqrt{2}}$

STEP 6

Simplify the expression by rationalizing the denominator:
$x = \frac{\sqrt{6}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$ $= \frac{\sqrt{12}}{2}$

STEP 7

Simplify $\sqrt{12}$ to find the final expression for $x$ :
$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$
Thus,
$x = \frac{2\sqrt{3}}{2} = \sqrt{3}$
The measure of $x$ is:
$x = \sqrt{3}$

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QuestionThis is a 45-45-90 triangle. What is the measure of xxx ? x=[?]x=\sqrt{[?]}x=[?]​

Studdy Solution

STEP 1

1. The triangle is a 45-45-90 triangle, meaning it is an isosceles right triangle.2. In a 45-45-90 triangle, the legs are equal in length, and the hypotenuse is x2 x\sqrt{2} x2​, where x x x is the length of each leg.

STEP 2

1. Understand the properties of a 45-45-90 triangle.2. Set up an equation based on the relationship between the legs and the hypotenuse.3. Solve for x x x.

STEP 3

Recognize that in a 45-45-90 triangle, the legs are equal, and the hypotenuse is x2 x\sqrt{2} x2​, where x x x is the length of each leg.

STEP 4

Set up the equation using the relationship of the hypotenuse to the legs in a 45-45-90 triangle. Given that the hypotenuse is 6\sqrt{6}6​, we have:x2=6 x\sqrt{2} = \sqrt{6} x2​=6​

STEP 5

Solve for x x x by dividing both sides of the equation by 2\sqrt{2}2​:x=62 x = \frac{\sqrt{6}}{\sqrt{2}} x=2​6​​

STEP 6

Simplify the expression by rationalizing the denominator:x=62×22 x = \frac{\sqrt{6}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} x=2​6​​×2​2​​ =122 = \frac{\sqrt{12}}{2} =212​​

STEP 7

Simplify 12\sqrt{12}12​ to find the final expression for x x x:12=4×3=4×3=23 \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} 12​=4×3​=4​×3​=23​Thus, x=232=3 x = \frac{2\sqrt{3}}{2} = \sqrt{3} x=223​​=3​ The measure of x x x is:x=3 x = \sqrt{3} x=3​

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QuestionThis is a 45-45-90 triangle. What is the measure of $x$ ? $x=\sqrt{[?]}$

1. The triangle is a 45-45-90 triangle, meaning it is an isosceles right triangle.
2. In a 45-45-90 triangle, the legs are equal in length, and the hypotenuse is $x\sqrt{2}$ , where $x$ is the length of each leg.

1. Understand the properties of a 45-45-90 triangle.
2. Set up an equation based on the relationship between the legs and the hypotenuse.
3. Solve for $x$ .

Recognize that in a 45-45-90 triangle, the legs are equal, and the hypotenuse is $x\sqrt{2}$ , where $x$ is the length of each leg.

Set up the equation using the relationship of the hypotenuse to the legs in a 45-45-90 triangle. Given that the hypotenuse is $\sqrt{6}$ , we have:
$x\sqrt{2} = \sqrt{6}$

Solve for $x$ by dividing both sides of the equation by $\sqrt{2}$ :
$x = \frac{\sqrt{6}}{\sqrt{2}}$

Simplify the expression by rationalizing the denominator:
$x = \frac{\sqrt{6}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$ $= \frac{\sqrt{12}}{2}$

Simplify $\sqrt{12}$ to find the final expression for $x$ :
$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$
Thus,
$x = \frac{2\sqrt{3}}{2} = \sqrt{3}$
The measure of $x$ is:
$x = \sqrt{3}$