The lengths of the diagnols of a rhombus are 2 in. and 5 in. Find the mesure of th angles of the rhombus.?

Question

I have no idea how to set this up...please help me solve this, I think this has something to do with the tangent of something...

Answer 1

Yes, tangent is involved. You aren't as clueless as you thought. And if you think about where tangent is used....
Right Triangles!
But what does that have to do with a rhombus?
Well, the diagonals of a rhombus are perpendicular bisectors of each other - they make right angles and cut each other in half. So, you have a right triangle with one leg = 1 and the other leg = 2.5. Take the inverse tangent of 2.5/1 - that's about 68.2 degrees. Double it as the diagonals bisect the opposite angles, 2*68.2 = 136.4 degrees are the measures of the two obtuse angles. The other angles are equal and have to make all 4 angles add up to 360, so those angles are each 43.6 degrees.

Answer 2

The area of a rhombus is half the product of its diagonals.

Area = (1/2)(d1)(d2) = (1/2)(2)(5) = 5

Let
s = length of side of rhombus

The diagonals of a rhombus always cross at right angles. When they do they form four congruent right triangles. The lengths of the two legs are half the length of each diagonal. The hypotenuse is one of the sides.

s² = (d1 / 2)² + (d2 / 2)² = (5/2)² + (2/2)² = 25/4 + 1 = 29/4

The area of a rhombus can also be expressed as:

Area = s² sinθ
where θ is the angle between two of the sides.

The acute angle plus the obtuse angle total 180° so the angles are θ and (180 - θ). Since the sine of either angle is the same it doesn't matter which angle you choose.

Area = s² sinθ
5 = s² sinθ
sinθ = 5/s² = 5 / (29/4) = 20/29

θ = arcsin(20/29) ≈ 43.6°

The obtuse angle is

180 - θ ≈ 136.4°

Answer 3

diagnols of rhombus intersect at 90 so

angle/2= tan inverse(2.5)

another angle/2=90-angle/2

Answer 4

p^2 + q^2 = 4a^2
2^2 + 5^2 = 4a^2
4a^2 = 4 + 25
4a^2 = 29
a^2 = 29/4
a = (1/2)sqrt(29)

p = a sqrt[2 - 2 cos(A)]
2 = (1/2)sqrt(29)sqrt(2 - 2cos(A))
4 = sqrt(29(2 - 2cos(A)))
16 = 29(2 - 2cos(A))
(16/29) = 2(1 - cos(A))
(8/29) = 1 - cos(A)
-cos(A) = -21/29
cos(A) = 21/29
A = about 43.6°

q = a sqrt[2 - 2 cos(B)]
5 = (sqrt(58(1 - cos(B)))/2
10 = sqrt(58(1 - cos(B)))
100 = 58(1 - cos(B))
(50/29) = 1 - cos(B)
-cos(B) = (21/29)
cos(B) = (-21/29)
B = about 136.4°

If you add the 2 exact values, you will get 180°, which the 2 angles are suppose to add up to be 180°.