A right-angled triangle has a catheter 7 cm shorter than the other catheter and a hypotenuse 6 cm longer. Determine the area of the triangle.
catheter? I don't think so.
The legs are x and x-7, and the hypotenuse is x+6. So,
x^2 + (x-7)^2 = (x+6)^2
Solve for x, and then of course, the area is 1/2 x(x-7)
Ok so.
x^2+(x-7)^2 = (x+6)^2
it eventually left me with:
=13±2√39
Is that right? Do I now have to that into 1/2 x(x-7) Or did I muck it up?
To solve this problem, we can use the properties of right-angled triangles and the Pythagorean theorem. Let's denote the length of one catheter as x cm.
According to the problem, the other catheter is 7 cm shorter than x, so its length would be (x - 7) cm.
The hypotenuse is 6 cm longer than x, so its length would be (x + 6) cm.
Now, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two other sides. In equation form, this is:
(x + 6)^2 = x^2 + (x - 7)^2
Expanding this equation, we get:
x^2 + 12x + 36 = x^2 + x^2 - 14x + 49
Combining like terms, we have:
2x^2 - 26x + 13 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Plugging in the values from our equation, we have:
x = (-(-26) ± sqrt((-26)^2 - 4(2)(13))) / (2(2))
Simplifying further:
x = (26 ± sqrt(676 - 104)) / 4
x = (26 ± sqrt(572)) / 4
Now, we evaluate two possible solutions for x:
x₁ = (26 + sqrt(572)) / 4 ≈ 8.07 cm
x₂ = (26 - sqrt(572)) / 4 ≈ 0.93 cm
Since we are dealing with physical lengths, the value x cannot be negative, so we discard the negative solution.
Now, we can use the value of x to find the lengths of the other sides:
The length of the second catheter = x - 7 ≈ 8.07 - 7 ≈ 1.07 cm
The length of the hypotenuse = x + 6 ≈ 8.07 + 6 ≈ 14.07 cm
Finally, we can calculate the area of the triangle using the formula for the area of a right-angled triangle:
Area = (1/2) * base * height
In this case, the base and height can be the two catheters, so:
Area = (1/2) * (x - 7) * x
Substituting the value of x, we have:
Area ≈ (1/2) * (8.07 - 7) * 8.07
Area ≈ (1/2) * 1.07 * 8.07
Area ≈ 4.33 square cm
Therefore, the area of the triangle is approximately 4.33 square cm.