Two legs of a right triangle are 3 inches and 4 inches long. What is the length of the longer leg of a similar right triangle if its shorter leg is 102 inches long?
3/4 = 102/x
or
x/4 = 102/3
To find the length of the longer leg of a similar right triangle, we can use the concept of proportionality.
The ratio of the lengths of the corresponding sides of two similar triangles is equal to the ratio of their corresponding sides. In this case, the ratio of the shorter legs is the same as the ratio of the longer legs.
Let's set up the proportion using the given lengths:
Shorter leg of the first triangle: 3 inches
Shorter leg of the second triangle: 102 inches
Longer leg of the first triangle: x inches (unknown)
Longer leg of the second triangle: y inches (unknown)
The proportion can be written as:
3 / x = 102 / y
To solve for y, we can cross-multiply:
3 * y = 102 * x
Now, let's substitute the known values:
3 * y = 102 * 4
Simplifying:
3 * y = 408
Finally, let's solve for y by dividing both sides of the equation by 3:
y = 408 / 3
y ≈ 136
Therefore, the length of the longer leg of the similar right triangle is approximately 136 inches.
To find the length of the longer leg of a similar right triangle, we can use the concept of proportions.
In a right triangle, the lengths of the legs are related by the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In mathematical terms, it can be written as:
a^2 + b^2 = c^2
where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse.
We are given the lengths of the two legs of the original right triangle: 3 inches and 4 inches. Let's call them a1 and b1, respectively.
a1 = 3 inches
b1 = 4 inches
Using the Pythagorean theorem, we can calculate the length of the hypotenuse c1:
c1^2 = a1^2 + b1^2
c1^2 = 3^2 + 4^2
c1^2 = 9 + 16
c1^2 = 25
Taking the square root of both sides, we get:
c1 = √25
c1 = 5 inches
Now, we can set up a proportion to find the length of the longer leg in the second right triangle. Let's call the length of the shorter leg a2 (which is given as 102 inches) and the length of the longer leg b2 (which we want to find).
So, the proportion is:
a1/a2 = b1/b2
Plugging in the values, we have:
3/102 = 4/b2
To solve for b2, we can cross-multiply:
3 * b2 = 4 * 102
3 * b2 = 408
b2 = 408 / 3
b2 = 136 inches
Therefore, the length of the longer leg of the similar right triangle is 136 inches.