The two legs of a right triangle are in the ratio √3/2. If the hypotenuse is 10 units long, find the area (in square units) of the triangle.
this is a 30-60-90 triangle. The short leg is half as long as the hypotenuse.
To find the area of the right triangle, we first need to determine the lengths of its legs. Let's represent the longer leg with the variable 'x' and the shorter leg with 'y.'
Given the ratio of the two legs as √3/2, we can set up the equation: x/y = √3/2
Cross-multiplying, we have x = (√3/2) * y
Since the hypotenuse is 10 units long, we can apply the Pythagorean Theorem to find the relationship between the lengths of the legs:
x^2 + y^2 = 10^2
Substituting x = (√3/2) * y into the equation:
(√3/2) * y^2 + y^2 = 100
Simplifying the equation:
(3/4) * y^2 + y^2 = 100
(7/4) * y^2 = 100
Multiplying both sides of the equation by 4/7:
y^2 = (100 * 4) / 7
y^2 = 400 / 7
Taking the square root of both sides:
y = √(400 / 7)
y = (√400) / (√7)
y = 20 / √7
Now that we have the length of one leg, we can find the length of the other leg by substituting this value into the equation x = (√3/2) * y:
x = (√3/2) * (20 / √7)
Simplifying the equation:
x = (√(3 * 20^2)) / (2 * √7)
x = (20√3) / (2√7)
x = 10√3 / √7
Thus, the lengths of the legs are x = 10√3 / √7 and y = 20 / √7.
To find the area of the triangle, we can use the formula A = (1/2) * base * height, where the base is one of the legs and the height is the other leg.
Let's take x as the base and y as the height:
A = (1/2) * (10√3 / √7) * (20 / √7)
Simplifying the equation:
A = (10 * 20 * √3) / (2 * √7 * √7)
A = (10 * 20 * √3) / (2 * 7)
A = (10 * 20 * √3) / 14
A = (200√3) / 14
A = (100√3) / 7
Therefore, the area of the triangle is (100√3) / 7 square units.
To find the area of a right triangle, we need to know the lengths of its two legs.
Given that the two legs of the right triangle are in the ratio √3/2, let's assume the lengths of the legs are (√3x/2) and x, where x is a constant.
We also know that the hypotenuse is 10 units long. Using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the two legs, we can set up the equation:
(√3x/2)^2 + x^2 = 10^2
Simplifying this equation, we have:
(3x^2)/4 + x^2 = 100
Multiplying both sides of the equation by 4 to clear the fraction, we get:
3x^2 + 4x^2 = 400
Combining like terms, we have:
7x^2 = 400
Now, let's solve for x by dividing both sides of the equation by 7:
x^2 = 400/7
Taking the square root of both sides, we find:
x = √(400/7)
Now, we have the value of x, which we can substitute back into our original assumption for the lengths of the legs. The length of one leg is (√3x/2) and the length of the other leg is x.
The area of a triangle can be calculated using the formula: Area = (1/2) * base * height.
In our case, the base is x and the height is (√3x/2).
Therefore, the area of the triangle is:
Area = (1/2) * x * (√3x/2)
Simplifying this expression, we have:
Area = (x * √3x) / 4
Now, substitute the value of x we found earlier:
Area = (√(400/7) * √3√(400/7)) / 4
Simplifying further, we get:
Area = (√(1200/49)) / 4
Taking the square root and simplifying:
Area = (20√3/7) / 4
Finally, dividing and simplifying, the area of the triangle is:
Area = 5√3 / 7
Therefore, the area of the triangle is 5√3 / 7 square units.