If the altitude of an isosceles right triangle has a length of x units, what is the length of one leg of the large right triangle in terms of [tex]$x$[/tex]?
A. [tex]$x$[/tex] units
B. [tex]$x \sqrt{2}$[/tex] units
C. [tex]$x \sqrt{3}$[/tex] units
D. [tex]$2 x$[/tex] units
Answer (2)
Define the isosceles right triangle with legs of length a and altitude x to the hypotenuse.
Calculate the area of the triangle in two ways: 2 1 a 2 and 2 1 ( a 2 ) x .
Equate the two area expressions: 2 1 a 2 = 2 1 ( a 2 ) x .
Solve for a in terms of x , which gives a = x 2 . The length of one leg of the triangle is x 2 .
Explanation
Problem Analysis Let's analyze the problem. We have an isosceles right triangle, which means it has two equal sides and a right angle. The altitude to the hypotenuse has a length of x units. We need to find the length of one of the legs of the triangle in terms of x .
Setting up the problem Let the length of each leg of the isosceles right triangle be a . Then the hypotenuse has length a 2 . The area of the triangle can be calculated in two ways: using the legs as base and height, or using the hypotenuse as base and the given altitude x as height.
Area using legs Using the legs as base and height, the area of the triangle is 2 1 ⋅ a ⋅ a = 2 1 a 2
Area using hypotenuse and altitude Using the hypotenuse as base and the altitude x as height, the area of the triangle is 2 1 ⋅ ( a 2 ) ⋅ x
Equating the areas Now we equate the two expressions for the area: 2 1 a 2 = 2 1 ( a 2 ) x
Simplifying the equation We can simplify this equation by multiplying both sides by 2: a 2 = a 2 x
Solving for a Now, divide both sides by a (since a cannot be zero): a = x 2
Final Answer Therefore, the length of one leg of the isosceles right triangle is x 2 units.
Examples
Isosceles right triangles are commonly used in construction and design. For example, if you are building a ramp that needs to have a specific height ( x ) and a 45-degree angle, you can use this relationship to determine the length of the base (which is also x 2 ) needed for the ramp.
The length of one leg of the isosceles right triangle in terms of the altitude x is x 2 units. The correct option is B. x 2 units.
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