Example Question - 45-45-90 triangle
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Calculation of Leg Length in 45-45-90 Triangle
This image depicts a right triangle, where one of the angles is 45 degrees and the hypotenuse opposite this angle measures 5 units. Since this is a 45-45-90 triangle, the two legs are congruent. In a 45-45-90 right triangle, the legs are each \( \frac{1}{\sqrt{2}} \) (which is the same as \( \sqrt{2}/2 \)) times the length of the hypotenuse. Let's call the length of each leg \( x \). Then: \[ x = \frac{1}{\sqrt{2}} \times 5 = \frac{5}{\sqrt{2}} \] However, it's often preferred to rationalize the denominator, so we multiply the numerator and denominator by \( \sqrt{2} \): \[ x = \frac{5\sqrt{2}}{\sqrt{2}\times\sqrt{2}} = \frac{5\sqrt{2}}{2} \] Therefore, the length of each leg of the triangle is \( \frac{5\sqrt{2}}{2} \) units.
Solving for the Length of Legs in a 45-45-90 Triangle
The image shows a right-angle triangle with a 45-degree angle. Since the angles in a triangle add up to 180 degrees and we already have a right angle (90 degrees) and one 45-degree angle, the other angle must also be 45 degrees. This makes the triangle a 45-45-90 triangle, which is a form of an isosceles right triangle. In a 45-45-90 triangle, the lengths of the legs (the two shorter sides) are equal, and the length of the hypotenuse (the longest side, opposite the right angle) is √2 times the length of a leg. The figure gives the hypotenuse as length 3. Therefore, to find the length of each leg (let's call it 'L'), we can use the proportion that L is to 3 (the hypotenuse) as 1 is to √2: L / 3 = 1 / √2 Multiplying both sides by 3 to solve for L: L = 3 / √2 To rationalize the denominator: L = (3 / √2) * (√2 / √2) L = (3√2) / 2 So, the length of each leg of the triangle is (3√2) / 2 units.
Solving for Legs of a 45-45-90 Triangle
The image shows a right triangle with one angle marked as 45 degrees and the hypotenuse labeled as 3 units long. This is an isosceles right angle triangle since one angle is 45 degrees, meaning that the other non-right angle is also 45 degrees because the two acute angles in a right triangle always add up to 90 degrees. In an isosceles right triangle, the legs (the two sides that are not the hypotenuse) are congruent, meaning they have the same length. To solve for the lengths of the legs, we can use the Pythagorean theorem or a knowledge of the properties of a 45-45-90 triangle. In a 45-45-90 triangle, the hypotenuse is √2 times longer than either of the legs. Let's call the length of each leg 'x'. According to the special properties of a 45-45-90 triangle: hypotenuse = leg * √2 Since the hypotenuse is 3 units, we can formulate the following equation: 3 = x * √2 To find 'x,' divide both sides of the equation by √2: x = 3 / √2 To rationalize the denominator, multiply the numerator and denominator by √2: x = (3 * √2) / (√2 * √2) x = (3 * √2) / 2 Therefore, the length of each leg of the triangle is 3√2/2 units.
Solving for Leg Lengths in a 45-45-90 Triangle
The image shows a right triangle with one angle of 45 degrees and the hypotenuse having a length of 3 units. Given that the triangle is a right triangle with a 45-degree angle, we can infer that this is a 45-45-90 triangle, which is an isosceles right triangle. In such triangles, the lengths of the legs are equal, and the length of the hypotenuse is √2 times the length of each leg. Let's use the properties of a 45-45-90 triangle to find the lengths of the two legs. If \( a \) is the length of one leg, we have: \( a \cdot \sqrt{2} = 3 \) Now we solve for \( a \): \( a = \frac{3}{\sqrt{2}} \) To rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{2} \): \( a = \frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \) \( a = \frac{3 \cdot \sqrt{2}}{2} \) So both legs of the triangle are \( \frac{3 \cdot \sqrt{2}}{2} \) units long.
Solving for 'x' in an Isosceles Right Triangle
It appears from the image that we have a right triangle with one of the angles being 45 degrees, which makes it an isosceles right triangle. In such a triangle, the two legs are congruent, which means they have the same length. The question asks for the value of "x" in each right angle. Since "x" is marked on the vertical leg of the triangle, and given the 45-degree angle, we can infer that "x" will also be the length of the horizontal leg because of the properties of an isosceles right triangle (45-45-90 triangle). Therefore, the value of "x" will be the same for both the vertical and horizontal legs of the triangle. In this type of triangle, the legs are equal in length, so x = x for both the horizontal and vertical legs. No numeric value can be provided without additional information.
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