The hypotenuse of a right triangle is 35cm and the perimeter is 84cm. Find the area of the triangle
Let the two legs of the right triangle be a and b.
By the Pythagorean theorem, we have:
a^2 + b^2 = 35^2.
Also, the perimeter of a triangle is the sum of all its sides. Thus, we have:
a + b + 35 = 84.
Rearranging the second equation, we get:
a + b = 84 - 35,
a + b = 49.
Solving the system of equations:
a + b = 49,
a^2 + b^2 = 35^2,
we can use the substitution method.
From the first equation, we obtain a = 49 - b.
Substituting this value of a into the second equation, we get:
(49 - b)^2 + b^2 = 35^2,
2401 - 98b + b^2 + b^2 = 1225,
2b^2 - 98b + 1176 = 0.
Dividing the equation by 2, we get:
b^2 - 49b + 588 = 0.
Factoring the equation, we have:
(b - 21)(b - 28) = 0.
Solving for b, we find two possible values:
b = 21 or b = 28.
If b = 21, substituting this value into the equation a + b = 49, we find:
a + 21 = 49,
a = 49 - 21,
a = 28.
If b = 28, substituting this value into the equation a + b = 49, we find:
a + 28 = 49,
a = 49 - 28,
a = 21.
In both cases, we obtain a = 21 and b = 28 as possible values of the legs.
Since the area of a right triangle is given by A = (1/2) * a * b, we can now calculate the area for each case:
For a = 21 and b = 28,
A = (1/2) * 21 * 28,
A = 294 square cm.
For a = 28 and b = 21,
A = (1/2) * 28 * 21,
A = 294 square cm.
Therefore, the area of the triangle is 294 square cm.
To find the area of a right triangle, we need to know the lengths of the two legs. However, in this case, we are given the length of the hypotenuse and the perimeter of the triangle.
Let's say the lengths of the two legs are x and y.
We are given:
Hypotenuse = 35 cm
Perimeter = 84 cm
The hypotenuse, in a right triangle, can be found using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.
Applying the Pythagorean theorem, we have:
x^2 + y^2 = 35^2
We are also given the perimeter of the triangle, which is the sum of the lengths of all three sides, so we can write:
x + y + hypotenuse = Perimeter
x + y + 35 = 84
Now, we have two equations with two unknowns. We can solve these equations simultaneously to find the values of x and y.
Let's solve the second equation for x:
x = 84 - y - 35
x = 49 - y
Substitute this value of x into the first equation:
(49 - y)^2 + y^2 = 35^2
Expanding and simplifying this equation:
2401 - 98y + y^2 + y^2 = 1225
Combine like terms:
2y^2 - 98y + 1176 = 0
Divide by 2 to simplify the equation:
y^2 - 49y + 588 = 0
Factorize the quadratic equation:
(y - 12)(y - 49) = 0
So, y = 12 or y = 49.
Substituting y = 12 into x = 49 - y, we get:
x = 49 - 12 = 37
Therefore, the lengths of the two legs are x = 37 cm and y = 12 cm.
To find the area of the triangle, we can use the formula for the area of a triangle:
Area = (base * height) / 2
In this case, the two legs of the triangle serve as the base and the height.
Area = (37 * 12) / 2
Area = 444 / 2
Area = 222 cm^2
So, the area of the triangle is 222 cm^2.
To find the area of the right triangle, we need to know the lengths of its two legs. However, we only know the hypotenuse and perimeter. So, let's assume the lengths of the two legs are a and b.
We know that the hypotenuse of a right triangle is always the longest side. In this case, the hypotenuse is given as 35 cm. Let's assume a < b, so a is one of the legs.
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs:
a^2 + b^2 = 35^2
We also know that the perimeter of a triangle is the sum of the lengths of all three sides:
a + b + Hypotenuse = 84
Substituting the value of the hypotenuse, we get:
a + b + 35 = 84
a + b = 84 - 35
a + b = 49
Now, we have two equations:
1. a^2 + b^2 = 35^2
2. a + b = 49
To solve these equations, we can use substitution or elimination method. Let's use substitution:
Rearrange the second equation to express a in terms of b:
a = 49 - b
Substitute this value of a in the first equation:
(49 - b)^2 + b^2 = 35^2
2401 - 98b + b^2 + b^2 = 1225
Combine like terms:
2b^2 - 98b + 1176 = 0
Now, we have a quadratic equation. We can solve it by factoring or using the quadratic formula. Factoring might not work in this case since the equation doesn't seem to have simple factors. So, let's use the quadratic formula:
b = (-(-98) ± √((-98)^2 - 4(2)(1176))) / (2(2))
Simplifying:
b = (98 ± √(9604 - 9408)) / 4
b = (98 ± √196) / 4
b = (98 ± 14) / 4
We get two potential values for b:
b1 = (98 + 14) / 4 = 112 / 4 = 28
b2 = (98 - 14) / 4 = 84 / 4 = 21
Now, substitute each value of b into the equation a + b = 49 to find the corresponding values of a:
When b = 28: a + 28 = 49 => a = 49 - 28 = 21
When b = 21: a + 21 = 49 => a = 49 - 21 = 28
We have two sets of values for a and b: (a = 21, b = 28) and (a = 28, b = 21). This is because we assumed a < b initially, but both solutions are valid.
Now, let's calculate the area of the triangle using the formula:
Area = (1/2) * base * height
We can use any of the two solutions. Let's use (a = 21, b = 28):
Area = (1/2) * 21 * 28 = 294 cm^2
Therefore, the area of the right triangle is 294 square cm.