The bisector of one of the angles of a rectangle also bisects a side of the rectangle. Find the perimeter of the rectangle if you know that the smaller side of the rectangle is equal to 10 cm.
I need the perimeter in cm please
you have to add 10 and 10 because the bisector splits side DC into two so instead of 20 you have 10, then AD is 10 so basically at the 20 and 20 and then the ten and ten so 20+20+10+10= 60 cm
Let's break down the problem step-by-step.
Step 1: Identify the given information.
- The smaller side of the rectangle is equal to 10 cm.
Step 2: Understand the problem.
- We are dealing with a rectangle.
- One of the angles of the rectangle is bisected by a line, which also bisects one of the sides.
Step 3: Find the lengths of the sides of the rectangle.
- Since the smaller side is given as 10 cm, we can assume that the opposite side is also 10 cm, as a rectangle has opposite sides of equal length.
- The line bisecting the angle and the side divides the other two sides of the rectangle into equal parts.
- Let's assume the length of one of these divided sides is 'x' cm.
Step 4: Determine the length of the remaining side.
- The remaining side of the rectangle can be found using the Pythagorean theorem, as we have a right-angled triangle formed by the divided side ('x') and the two 10 cm sides.
- Applying the theorem, we have:
x^2 + 10^2 = 10^2
x^2 = 10^2 - 10^2
x^2 = 100 - 100
x^2 = 0
x = 0
Step 5: Analyze the result.
- We found that the length of one of the divided sides is 0 cm.
- This suggests that the bisector does not divide the side into two equal parts.
Step 6: Answer the question.
- Since we couldn't find a length for one of the divided sides, it is not possible to determine the perimeter of the rectangle with the given information.
Therefore, the perimeter of the rectangle cannot be determined.
To find the perimeter of the rectangle, we need to determine the lengths of all four sides. Let's denote the longer side as 'L' and the shorter side as 'W'.
Given that the smaller side is 10 cm, we can establish that W = 10 cm.
Since the bisector of one of the angles also bisects a side of the rectangle, we can conclude that the rectangle is symmetrical along that bisector. This means that the remaining length of the shorter side is also 10 cm, making W = 10 cm on both sides.
Now, to find the length of the longer side, we'll use the fact that the bisector of one angle divides the rectangle into two congruent right triangles. The hypotenuse of each right triangle corresponds to the longer side of the rectangle.
Since the two right triangles are congruent, their corresponding sides are equal. Let's denote the length of the longer side, or the hypotenuse of each right triangle, as 'L'.
Using the Pythagorean theorem, the longer side can be determined as follows:
L^2 = (10 cm)^2 + (10 cm)^2
L^2 = 100 cm^2 + 100 cm^2
L^2 = 200 cm^2
L = √(200 cm^2)
L ≈ 14.14 cm (rounded to two decimal places)
Now that we know both the lengths of the shorter side (W = 10 cm) and the longer side (L ≈ 14.14 cm), we can calculate the perimeter of the rectangle.
Perimeter = 2W + 2L
Perimeter = 2(10 cm) + 2(14.14 cm)
Perimeter ≈ 20 cm + 28.28 cm
Perimeter ≈ 48.28 cm
Therefore, the perimeter of the rectangle is approximately 48.28 cm.