The diagonals of a rhombus are 10 and 24. Find the perimeter. Draw a picture to get started. Show all necessary steps.
Understand the Problem
The question asks for the perimeter of a rhombus given the lengths of its diagonals, which are 10 and 24. It suggests drawing a picture to aid in the solution and requires showing all necessary steps.
Answer
The perimeter of the rhombus is \( 52 \).
Answer for screen readers
The perimeter of the rhombus is ( P = 52 ).
Steps to Solve
- Understand the properties of a rhombus
In a rhombus, the diagonals bisect each other at right angles. Given diagonals of lengths 10 and 24, each half of the diagonals can be calculated:
- Half of the first diagonal: $\frac{10}{2} = 5$
- Half of the second diagonal: $\frac{24}{2} = 12$
- Calculate the sides of the rhombus
Using the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse, we can find the length of each side of the rhombus: $$ s = \sqrt{a^2 + b^2} $$ where $a$ and $b$ are half of the diagonals.
Here, $a = 5$ and $b = 12$. Thus, we calculate: $$ s = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 $$
- Calculate the perimeter of the rhombus
The perimeter $P$ of a rhombus is given by: $$ P = 4s $$ Substituting the side length: $$ P = 4 \times 13 = 52 $$
The perimeter of the rhombus is ( P = 52 ).
More Information
The perimeter calculation relies on the properties of a rhombus, utilizing the Pythagorean theorem to find the side lengths from the diagonals. Since the lengths of the diagonals divide the rhombus into four right triangles, this approach is effective.
Tips
- Forgetting to halve the diagonals: Each diagonal must be split into its two halves before applying the Pythagorean theorem.
- Misapplying the Pythagorean theorem: Ensure to correctly identify the right triangle formed by the half diagonals to solve for the side length.
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