Each side of a rhombus is 5 inches long and the shorter diagonal is 4 inches. Find the angle of the rhombus and the longer diagonal.
Use the fact that the diagonals of a rhombus right-bisect each other, so you will have 4 identical right-angled triangles.
Use x^2 + 2^2 = 5^2 to find half of the longer diagonal.
Use cosØ = 2/5 to find the base angle Ø of the triangle.
To find the angle of the rhombus, we need to use the formula for the interior angles of a rhombus. The formula is given by:
angle = arccos((a^2 + b^2 - c^2) / (2ab))
Where:
- angle represents the measure of the angle in radians,
- a and b represent the lengths of the sides of the rhombus,
- c represents the length of the shorter diagonal.
In this case, a = b = 5 inches and c = 4 inches.
So, plugging in the values, we have:
angle = arccos((5^2 + 5^2 - 4^2) / (2 * 5 * 5))
Simplifying further:
angle = arccos((25 + 25 - 16) / 50)
angle = arccos(34/50)
Using a calculator or math software, we can find the approximate value of arccos(34/50) to be 0.895 radians.
To find the longer diagonal, we can use the property of a rhombus that states that the diagonals of a rhombus are perpendicular bisectors of each other.
First, we need to find the length of the longer diagonal using the Pythagorean theorem. Let's call the longer diagonal d.
The Pythagorean theorem states:
d^2 = (a/2)^2 + c^2
Plugging in the values, we have:
d^2 = (5/2)^2 + 4^2
Simplifying:
d^2 = 6.25 + 16
d^2 = 22.25
Taking the square root of both sides:
d = √22.25
Using a calculator or math software, we can find the approximate value of √22.25 to be 4.714 inches.
Therefore, the angle of the rhombus is approximately 0.895 radians, and the longer diagonal is approximately 4.714 inches.