A rhombus of side 10cm has obtuse angles 110 degrees. Sketch the rhombus, showing it’s diagonals and as many angles as possible. Hence calculate the lengths of the diagonals of the rhombus

Answer 1

To sketch the rhombus, we start by drawing a horizontal line of length 10 cm. We then draw a line segment of length 5 cm at an angle of 55 degrees from each end of the horizontal line, going upwards. We then connect the endpoints of these two line segments with a horizontal line of length 10 cm, completing the rhombus.

To show the diagonals, we can draw a line from one corner of the rhombus to the opposite corner. This diagonal will have a length equal to the hypotenuse of a right triangle with legs of 5 cm (half the side length) and 10 cm (the distance between opposite corners). Using the Pythagorean theorem, we can calculate the length of the diagonal:

diagonal^2 = 5^2 + 10^2
diagonal^2 = 25 + 100
diagonal^2 = 125
diagonal = √125 ≈ 11.2 cm

We can repeat this process for the other diagonal:

diagonal^2 = 5^2 + 10^2
diagonal^2 = 25 + 100
diagonal^2 = 125
diagonal = √125 ≈ 11.2 cm

Therefore, the length of both diagonals of the rhombus is approximately 11.2 cm.

To indicate the obtuse angles on the sketch, we can label them using the measure of the angle in degrees. For example, we can label the obtuse angles as 110 degrees, as stated in the problem. If we extend the diagonals to their intersection point, we can also label the angles formed by the diagonals (which are all acute angles).

Answer 2

Not sure where you're getting these legs of length 5 cm.
The diagonals of the rhombus for four congruent right triangles; the sides of the rhombus are the hypotenuses of these right triangles. If the diagonals have length 2x and 2y, then
x = 10 cos55°
y = 10 cos35°
That means the diagonals have length 16.38 and 11.47

Answer 3

I apologize for the mistake in my previous response. You are correct that the diagonals of the rhombus are equal to twice the cosine of the acute angles formed by the diagonals.

Given that the obtuse angle of the rhombus is 110 degrees, we know that the acute angles formed by the diagonals are each half of the remaining angle, which is:

180 - 110 = 70 degrees

Using the formulas you provided,

x = 10 cos(55°) ≈ 8.09 cm
y = 10 cos(35°) ≈ 8.66 cm

Therefore, the length of each diagonal is:

2x ≈ 16.18 cm
2y ≈ 17.32 cm

So the length of both diagonals of the rhombus are approximately 16.18 cm and 17.32 cm.