If the area of triangle A with sides 5, 12, and 13 equals the area of rectangle B with height 5, what is the perimeter of the rectangle ?
5, 12, 13 is Pythagorean triple ... right triangle with integer sides
... the area is ... 1/2 * 5 * 12 = 30
for the rectangle ... 5 * w = 30
solve for w , then find the perimeter ... 5 + 5 + w + w
Why did the triangle want to hang out with the rectangle? Because they were both really into shapes!
To find the perimeter of the rectangle, we need to know its length. Unfortunately, the length is not given in the question. Can you please provide the length of the rectangle?
To find the perimeter of a rectangle, we need to know the lengths of its sides. In this case, we are given that the height of rectangle B is 5.
Since the area of triangle A is the same as the area of rectangle B, we can use the formula for the area of a triangle to calculate the height of triangle A.
The formula for the area of a triangle is: Area = (base * height) / 2.
Let's calculate the height of triangle A first:
Area of triangle A = Area of rectangle B
(5 * height of triangle A) / 2 = 5 [since the height of rectangle B is given as 5]
Simplifying the equation, we have:
5 * height of triangle A = 10
height of triangle A = 10 / 5
height of triangle A = 2
Now, we can calculate the length of the base of triangle A using the Pythagorean theorem since sides 5, 12, and 13 form a Pythagorean triple:
5^2 + 12^2 = 13^2
25 + 144 = 169
169 = 169
So, the length of the base of triangle A is 12.
Now that we know the height and base of triangle A, we can calculate its area:
Area of triangle A = (base * height) / 2
Area of triangle A = (12 * 2) / 2
Area of triangle A = 12
Since the area of triangle A equals the area of rectangle B and the height of rectangle B is 5, we can calculate the width of the rectangle:
Area of rectangle B = width * height
12 = width * 5
width = 12 / 5
Now, we can calculate the perimeter of rectangle B:
Perimeter of rectangle B = 2 * (width + height)
Perimeter of rectangle B = 2 * (12/5 + 5)
Perimeter of rectangle B = 2 * (12/5 + 25/5)
Perimeter of rectangle B = 2 * (37/5)
Perimeter of rectangle B = 74/5
So, the perimeter of rectangle B is 14.8 units.
To solve this problem, we need to use the formulas for the area of a triangle and the area of a rectangle.
The formula for the area of a triangle is given by:
Area = (1/2) * base * height
The formula for the area of a rectangle is given by:
Area = length * width
Let's start by finding the area of triangle A. We know that the sides of the triangle are 5, 12, and 13. To find the area, we can use Heron's formula, which states that for a triangle with sides a, b, and c, the area is given by:
Area = √(s * (s - a) * (s - b) * (s - c))
where s is the semi-perimeter of the triangle, given by:
s = (a + b + c) / 2
In this case, the sides of triangle A are 5, 12, and 13. So, let's calculate the semi-perimeter first:
s = (5 + 12 + 13) / 2
= 15
Now, we can find the area of triangle A:
Area = √(15 * (15 - 5) * (15 - 12) * (15 - 13))
= √(15 * 10 * 3 * 2)
= √(900)
= 30
Next, we'll find the area of rectangle B. We are given that the height of the rectangle is 5. The area is given by:
Area = height * width
Since we are looking for the perimeter of the rectangle, we need to find the width. We can rearrange the formula for the area to solve for the width:
width = Area / height
In this case, the area of rectangle B is equal to the area of triangle A, which is 30. The height is given as 5. So, the width of the rectangle is:
width = 30 / 5
= 6
Finally, to find the perimeter of the rectangle, we use the formula:
Perimeter = 2 * (length + width)
Since we have the height as 5 and the width as 6, the length would be equal to the height as the opposite sides of a rectangle are equal:
Perimeter = 2 * (5 + 6)
= 2 * 11
= 22
Therefore, the perimeter of the rectangle B is 22 units.