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. Please provide COMPLETE solution and FINAL ANSWER.
the sides would be √3x and 2x , then
(√3x)^2 + (2x)^2 = 10^2
3x^2 + 4x^2 = 100
solve for x and you have the two sides.
area = (1/2)(side1)(side2)
= ....
Hmmm. I see I misspoke before. I thought the sides were in the ratio √3.
So, let's go with what they gave you. If one side is 2x, then the other side is √3x
Now recall your Pythagorean Theorem. It says that
(2x)^2 + (√3x)^2 = 10^2
4x^2 + 3x^2 = 100
I know you want to just sit and wait for the answer, but I'm sure you can take it from here. One you have the two legs, it's easy to get the area.
Let's assume that the shorter leg of the right triangle is represented by "x" units.
According to the given information, the longer leg can be represented as √3/2 * x.
Now, we can use the Pythagorean theorem to find the lengths of the legs:
x^2 + (√3/2 * x)^2 = 10^2
Simplifying the equation:
x^2 + (3/4 * x^2) = 100
Combining like terms:
(1 + 3/4) * x^2 = 100
Multiplying:
(7/4) * x^2 = 100
Now divide both sides of the equation by (7/4) to isolate x^2:
x^2 = (100 * 4) / 7
x^2 = 400 / 7
To find the value of x, take the square root of both sides:
x = √(400 / 7)
x ≈ 8.92 units (rounded to two decimal places)
Now we can find the longer leg:
Longer leg = (√3/2 * 8.92) ≈ 6.13 units (rounded to two decimal places)
To find the area of the triangle, use the formula:
Area = (base * height) / 2
Area = (8.92 * 6.13) / 2
Area ≈ 27.33 square units (rounded to two decimal places)
Therefore, the area of the triangle is approximately 27.33 square units.
To find the area of a right triangle, we need the lengths of its two legs. Let's call the length of one leg x and the length of the other leg y.
Given that the two legs are in the ratio √3/2, we can write a proportion:
x/y = √3/2
To solve for x and y, we need additional information. In this case, we know that the hypotenuse is 10 units long. Using the Pythagorean theorem, we can relate the lengths of the legs and the hypotenuse:
x^2 + y^2 = 10^2
Now, let's solve the proportion for x:
x/y = √3/2
Cross multiplying, we have:
2x = y√3
Squaring both sides, we get:
4x^2 = 3y^2
By substitution, we can replace y^2 with (10^2 - x^2), our equation from the Pythagorean theorem:
4x^2 = 3(10^2 - x^2)
Expanding, we have:
4x^2 = 300 - 3x^2
Simplifying, we get:
7x^2 = 300
Dividing both sides by 7, we find:
x^2 = 300/7
Taking the square root of both sides, we have:
x = √(300/7)
To calculate the value of x, we can simplify the expression √(300/7) on a calculator or computer, which gives us x ≈ 8.51.
Now, let's find the value of y using the proportion:
x/y = √3/2
Substituting the value of x, we have:
8.51/y = √3/2
Cross multiplying, we get:
17.02 = √3y
Squaring both sides, we have:
289.0404 = 3y^2
Dividing both sides by 3, we find:
y^2 ≈ 96.3468
Taking the square root, we have:
y ≈ √96.3468 ≈ 9.82
Now, we have the lengths of the two legs: x ≈ 8.51 and y ≈ 9.82.
To find the area of the triangle, we use the formula:
Area = (1/2) * base * height
In this case, the base and height are the lengths of the two legs. So, the area is:
Area = (1/2) * 8.51 * 9.82 ≈ 41.96 square units.
Therefore, the area of the triangle is approximately 41.96 square units.