The hypotenuse of a right triangle os 13.0 and one leg is 2.0 units shorter than the other. Find the dimensions of the figure (rounded to the nearest tenth).

longer of the 2 legs .... x

shorter of the 2 legs = x-2

x^2 + (x-2)^2 = 13^2
x^2 + x^2 - 4x + 4 = 169
2x^2 - 4x - 165 = 0

use the quadratic formula to find x, reject the negative answer of x

Well, well, well, looks like we have ourselves a right triangle party! Let's unravel this puzzle.

We know that the hypotenuse of the triangle is 13.0 units, and one leg is 2.0 units shorter than the other. So, let's call the longer leg x units (because x is just so hip), and the shorter leg (x - 2) units (because it likes to be different).

Now, let's put our thinking cap on and recall a very special friend named Pythagoras. According to him, the sum of the squares of the legs will be equal to the square of the hypotenuse. That's a fancy way of saying x^2 + (x - 2)^2 = 13^2.

So, let's simplify that equation and solve it like a boss:
x^2 + x^2 - 4x + 4 = 169

2x^2 - 4x - 165 = 0

Using the quadratic formula, we find that x ≈ 14.7 or x ≈ -5.6. But we're talking about lengths here, so negative values are not our jam.

Therefore, the longer leg is approximately 14.7 units and the shorter leg is approximately (14.7 - 2) = 12.7 units.

Round those bad boys to the nearest tenth, and voila! The dimensions of our triangle are approximately 14.7 and 12.7 units.

So there you have it, the mystery of the right triangle is solved! Keep up the good work, triangle wrangler!

Let's call one leg of the right triangle "x" units. Since the other leg is 2.0 units shorter, we can represent it as "x - 2" units.

According to the Pythagorean theorem, the sum of the squares of the two legs is equal to the square of the hypotenuse.

So we have the equation: x^2 + (x - 2)^2 = 13^2

Expanding the equation, we get: x^2 + x^2 - 4x + 4 = 169

Combining like terms, we have: 2x^2 - 4x - 165 = 0

To solve for x, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a

For our equation, a = 2, b = -4, and c = -165. Substituting these values into the quadratic formula, we get:

x = (-(-4) ± √((-4)^2 - 4(2)(-165))) / (2(2))
x = (4 ± √(16 + 1320)) / 4
x = (4 ± √(1336)) / 4

Now, we can simplify the equation:

x ≈ (4 ± 36.57) / 4

Calculating both possible values for x, we have:

x ≈ (4 + 36.57) / 4 ≈ 10.64 / 4 ≈ 2.7
and
x ≈ (4 - 36.57) / 4 ≈ -32.57 / 4 ≈ -8.1

Since lengths cannot be negative, we can conclude that the value of x is approximately 2.7.

Therefore, one leg of the triangle is 2.7 units and the other leg is 2 units shorter, so it is approximately 0.7 units.

Rounded to the nearest tenth, the dimensions of the triangle are 2.7 units and 0.7 units.

To find the dimensions of the right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

Let's assume one leg of the triangle is equal to x units. Since the other leg is 2 units shorter, the length of the other leg is equal to x - 2 units.

Using the Pythagorean theorem, we can set up the following equation:

(x - 2)^2 + x^2 = 13^2

Expanding the equation:

x^2 - 4x + 4 + x^2 = 169

Combining like terms:

2x^2 - 4x + 4 = 169

Rearranging the equation to standard quadratic form:

2x^2 - 4x - 165 = 0

Now, we can solve this quadratic equation for x. We can use the quadratic formula:

x = (-b ± √(b^2 - 4ac))/(2a)

In this case, a = 2, b = -4, and c = -165. Plugging in these values, we have:

x = (-(-4) ± √((-4)^2 - 4(2)(-165))) / (2(2))

Simplifying:

x = (4 ± √(16 + 1320)) / 4

x = (4 ± √1336) / 4

x = (4 ± 36.56) / 4

Now, we have two possible solutions for x:

1) x = (4 + 36.56) / 4 = 40.56 / 4 = 10.14
2) x = (4 - 36.56) / 4 = -32.56 / 4 = -8.14

Since the dimensions of a triangle cannot be negative, we can discard the second solution, leaving us with x = 10.14.

Therefore, the two legs of the right triangle are approximately 10.14 units and 8.14 units long (10.14 - 2). Rounded to the nearest tenth, the dimensions are 10.1 units and 8.1 units.