the sum of area and perimeter ofintegral right angle triangle is 288. find the sides.
check the easy right triangles:
3,4,5: perimeter = 12, which is 288/24, so
scale by 24 to get 72,96,120
total: 288
sorry, I missed the area part.
1/2 bh + b+h+√(b^2+h^2) = 288
check 3,4,5:
scale by n, and p=12n, a=6n^2, p+a=6n(n+2)
288/6 = 48 = 6*8, so n = 6
scale 3,4,5 by 6 to get
18,24,30
p=72
a=216
p+a=288
Let's assume the sides of the integral right-angled triangle are represented by the variables 'a', 'b', and 'c'. The hypotenuse is denoted by 'c', and the perpendicular and base are represented by 'a' and 'b' respectively.
The formulas for the area and perimeter of a right-angled triangle are as follows:
Area = (1/2) * base * height
Perimeter = base + height + hypotenuse
Given that the sum of the area and perimeter is 288, we can write the equation:
Area + Perimeter = 288
(1/2) * a * b + a + b + c = 288
Now let's solve this equation step-by-step:
1. Rearrange the equation to isolate 'c':
(1/2) * a * b + a + b + c = 288
c = 288 - [(1/2) * a * b + a + b]
2. Substitute the Pythagorean theorem into the equation:
a^2 + b^2 = c^2
c = sqrt(a^2 + b^2)
Substituting back into the previous equation:
c = 288 - [(1/2) * a * b + a + b]
3. Substitute the value of 'c' in terms of 'a' and 'b':
288 - [(1/2) * a * b + a + b] = sqrt(a^2 + b^2)
Square both sides to eliminate the square root:
(288 - [(1/2) * a * b + a + b])^2 = a^2 + b^2
4. Simplify and rearrange the equation:
(288 - (1/2) * a * b - a - b)^2 = a^2 + b^2
Simplify the left-hand side using the distributive property:
(288 - [1/2 * a * b + a + b])^2 = a^2 + b^2
Expand and simplify the left-hand side:
(288 - a - b - [1/2 * a * b])^2 = a^2 + b^2
Expand the squared term:
(288^2 + a^2 + b^2 + 2 * 288 * (-a) + 2 * 288 * (-b) + 2 * 288 * (-[1/2 * a * b]) + (-a)^2 + (-b)^2 + ([1/2 * a * b])^2) = a^2 + b^2
Simplify and cancel terms:
288^2 + 2 * 288 * (-a) + 2 * 288 * (-b) + 2 * 288 * (-[1/2 * a * b]) + (-a)^2 + (-b)^2 + ([1/2 * a * b])^2 = 0
5. Rearrange the equation and cancel terms:
288^2 - 2 * 288 * a - 2 * 288 * b - a^2 - b^2 + ([1/2 * a * b])^2 = 0
6. Simplify and gather like terms:
82944 - 576 * a - 576 * b - a^2 - b^2 + ([1/4 * a^2 * b^2]) = 0
7. Simplify further:
[a^2 + (576 * a) + b^2 + (576 * b)] - 82944 + ([1/4 * a^2 * b^2]) = 0
Now, you could use numerical methods or algebraic techniques to solve this equation. Unfortunately, there isn't enough information given to find the specific values for 'a', 'b', and 'c' without further constraints.
To find the sides of the integral right-angled triangle, we need to set up and solve equations based on the given information.
Let's denote the two shorter sides of the triangle as 'a' and 'b' (with a ≤ b), and the hypotenuse as 'c'.
The area of a right-angled triangle is given by: Area = (1/2) × (a × b)
The perimeter (sum of all sides) is given by: Perimeter = a + b + c
From the given information, we know that the sum of the area and perimeter of the triangle is 288. Thus, we can formulate the equation:
(1/2) × (a × b) + a + b + c = 288
Since it is an integral right-angled triangle, a, b, and c should all be positive integers.
To simplify the equation, we can multiply both sides by 2:
a × b + 2a + 2b + 2c = 576
Rearranging the terms, we get:
a × b + 2(a + b + c) = 576
Now, let's focus on finding all the possible values of 'a', 'b', and 'c' that satisfy this equation:
1. Start by trying different values of 'a' and 'b' such that a times b equals 576.
For 'a' = 1: The factors of 576 are (1, 576), (2, 288), (3, 192), (4, 144), (6, 96), (8, 72), (9, 64), (12, 48), (16, 36), (18, 32), (24, 24).
Take the first pair (1, 576) and see if it satisfies the equation.
1 × 576 + 2(1 + 576 + c) = 576 + 2(577 + c) = 576 + 1154 + 2c = 1730 + 2c ≠ 576
This pair doesn't satisfy the equation, so move on to the next pair.
For 'a' = 2: Trying all the above factor pairs, we find that (2, 288) satisfies the equation.
2 × 288 + 2(2 + 288 + c) = 576 + 2(290 + c) = 1156 + 2c = 1730 + 2c = 288
We have found a solution: a = 2, b = 288, c = 290.
So, the sides of the integer right-angled triangle are 2, 288, and 290.
Note: It's possible that there could be other solutions, so it's always a good idea to check all possible factor pairs of 576.