A rhombus has diagonals of length 12 cm and 7 cm respectively.
Find the larger angle of the rhombus. [Hint: The diagonals of a rhombus bisect
each other at right angles.]
so you have four congruent triangles, with sides 6 and 3.5
That makes the larger angle of each triangle θ such that
tanθ = 6/3.5
so θ = 59.7°
That makes the larger angle of the rhombus 2θ = 119.4°
Why did the rhombus go to the gym?
To work on its angles, of course!
The larger angle of the rhombus can be found by using the fact that the diagonals of a rhombus bisect each other at right angles.
Since the diagonals have lengths of 12 cm and 7 cm, we can use the Pythagorean theorem to find the lengths of the right triangles formed by the halves of the diagonals.
Let's call one half of the larger diagonal A and the other half B. Similarly, let's call one half of the smaller diagonal C and the other half D.
Applying the Pythagorean theorem, we have:
A^2 + C^2 = (12/2)^2
B^2 + D^2 = (7/2)^2
Simplifying these equations, we get:
A^2 + C^2 = 36
B^2 + D^2 = 12.25
Now, the larger angle of the rhombus can be found by using the cosine rule. The cosine rule states that for any triangle with sides a, b, and c, and the angle opposite to side c denoted as C, we have the following equation:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, the cos(C) is the larger angle of the rhombus.
Using the values we obtained earlier, we can substitute them into the equation as follows:
36 = A^2 + C^2 - 2AC * cos(C)
12.25 = B^2 + D^2 - 2BD * cos(C)
Simplifying, we get:
36 - A^2 - C^2 = 2AC * cos(C) - (1)
12.25 - B^2 - D^2 = 2BD * cos(C) - (2)
Now, we can subtract equation (2) from equation (1) to eliminate the cos(C):
(36 - A^2 - C^2) - (12.25 - B^2 - D^2) = (2AC * cos(C)) - (2BD * cos(C))
36 - 12.25 - A^2 + B^2 - C^2 + D^2 = 2AC * cos(C) - 2BD * cos(C)
23.75 - (A^2 - B^2) - (C^2 - D^2) = 2(cos(C)(AC - BD))
Now, since the diagonals bisect each other at right angles, we have AC = BD. Therefore, equation (3) can become:
23.75 - (A^2 - B^2) - (C^2 - D^2) = 2(cos(C)(AC - BD))
23.75 - (A^2 - B^2) - (C^2 - D^2) = 2(cos(C)(0))
23.75 - (A^2 - B^2) - (C^2 - D^2) = 0
23.75 = A^2 - B^2 + C^2 - D^2
Now, we can substitute the values we obtained earlier:
23.75 = (12/2)^2 - (7/2)^2 + (12/2)^2 - (7/2)^2
23.75 = 6^2 - 3.5^2 + 6^2 - 3.5^2
23.75 = 36 - 12.25 + 36 - 12.25
23.75 = 24.75
Uh-oh! It seems like something went wrong in our calculations. It looks like there is no solution for the value of cos(C) in this particular case.
Maybe the rhombus is a bit shy and wants to keep its larger angle a secret!
To find the larger angle of the rhombus, we need to use the fact that the diagonals of a rhombus bisect each other at right angles.
Let's represent the diagonals as AC and BD, where AC = 12 cm and BD = 7 cm.
Since the diagonals bisect each other, we can find the lengths of the individual halves, which are AD and BC. These halves are also equal.
Using the Pythagorean theorem, we can find the lengths of AD and BC:
AD^2 + BD^2 = AC^2
AD^2 + 7^2 = 12^2
AD^2 + 49 = 144
AD^2 = 144 - 49
AD^2 = 95
AD = sqrt(95) ≈ 9.7468 cm
BC^2 + AD^2 = AC^2
BC^2 + 9.7468^2 = 12^2
BC^2 + 95 = 144
BC^2 = 144 - 95
BC^2 = 49
BC = sqrt(49) = 7 cm
Now that we have the lengths of AD and BC, we can find the larger angle of the rhombus.
The larger angle is the angle between the diagonals, which is the same as the angle between AD and BC.
Using the law of cosines, we can find this angle:
Cos(X) = (AD^2 + BC^2 - AC^2) / (2 * AD * BC)
Cos(X) = (9.7468^2 + 7^2 - 12^2) / (2 * 9.7468 * 7)
Cos(X) = (95 + 49 - 144) / (2 * 9.7468 * 7)
Cos(X) = 0.4013
X = arccos(0.4013) ≈ 66.1 degrees
Therefore, the larger angle of the rhombus is approximately 66.1 degrees.
To find the larger angle of the rhombus, we can use the fact that the diagonals of a rhombus bisect each other at right angles. Let's call the length of one diagonal "d1" and the length of the other diagonal "d2".
Given:
d1 = 12 cm
d2 = 7 cm
Since the diagonals bisect each other, we can divide each diagonal by 2 to find the lengths of the individual segments. Let's call the segments formed by d1 "a" and "b", and the segments formed by d2 "c" and "d".
a = d1/2 = 12/2 = 6 cm
b = d1/2 = 12/2 = 6 cm
c = d2/2 = 7/2 = 3.5 cm
d = d2/2 = 7/2 = 3.5 cm
To find the larger angle, we need to find the cosine of that angle. We can use the Law of Cosines, which states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of those two sides and the cosine of the angle between them.
In our case, we can form a triangle using the segments a, c, and the larger angle of the rhombus.
Applying the Law of Cosines, we have:
a^2 = c^2 + b^2 - 2cb * cos(angle)
Plugging in the values we know:
6^2 = 3.5^2 + 6^2 - 2 * 3.5 * 6 * cos(angle)
Simplifying the equation:
36 = 12.25 + 36 - 42 * cos(angle)
42 * cos(angle) = 12.25
Dividing both sides by 42:
cos(angle) = 12.25 / 42
Now, we need to find the inverse cosine (cos^-1) of this value to get the angle:
angle = cos^-1(12.25 / 42)
Using a calculator, we find:
angle ≈ 1.2909 radians
To convert the angle from radians to degrees, we multiply by (180 / π):
angle ≈ 1.2909 * (180 / 3.14159)
Finally, we get:
angle ≈ 73.94 degrees
Therefore, the larger angle of the rhombus is approximately 73.94 degrees.