Trig Help Please?

enter image source here (a) Find length of a side of the rhombus to the nearest tenth.
(b) Using your answer in part a, find the perimeter of rhombus ABCD.
(c) Find the length of diagonal (BD)to the nearest integer.

2 Answers
Jan 27, 2018

a) sides are all 49.449.4

b) Perimeter = 197.6=197.6

c) BD = 58BD=58

Explanation:

Let's call the intersection of the diagonals point M,

Remember that in rhombus the diagonals bisect each other and intersect at 90°, so we have right angled triangles to work with.

In DeltaBAM we have: hatA =36°, hatM = 90° and AM = 40

a) With reference to hatA, AM is the adjacent side and color(blue)(AB) is the hypotenuse which is the length we need to know.

We can therefore use the Cos ratio ("adjacent"/"hypotenuse")

Cos36°=(AM)/(color(blue)(AB)) = 40/(AB)

color(blue)(AB) xxcos36° = 40

color(blue)(AB) = 40/cos36°

color(blue)(AB)= 49.4

b) In a rhombus all the sides are equal, so now that we know the length of AB, the perimeter is 4 xx AB

P = 4 xx 49.4 = 197.6

c) You can either use Pythagoras'theorem or trig to find the length of BM which will be doubled to give BD

BM^2 = 49.4^2-40^2

BM^2 = 840.36

BM = sqrt840.36

BM = 28.98896

BD = 2 xx 28.98896

BD = 58

Using trig is slightly more accurate because we use the original values, not arounded off value.

(BM)/(AM) = tan36°

BM = 40tan36°

BM = 29.06

BD = 2 xx 29.06 = 58

Jan 27, 2018

(a) Side approx 49.4; (b) Perimeter approx 197.8; (c) Diagonal BD approx 58

Explanation:

Using diagram in question.

Since ABCD is a rhombus -> AB=BC=CD=AD

(a) :.triangle ABC is isoceles -> angle BCA = 36^o

:. angle ABC = 180 -72 = 108^o

Applying the sine rule to triangle ABC

80/sin108 = (BC)/sin 36

BC = (80sin36)/sin108

approx 49.44272 = 49.4 (1D)

(b) The four sides of a rhombus are equal.

:. Perimeter approx 49.44272 xx4 = 197.8 (1D)

(c) Diagonal BD and diagonal AC bisect eachother at 90^o

Let the point of intersection be O

:. triangle BOC is a right triangle.

Apply Pythagoras

BO^2 + OC^2 = BC^2

BO^2approx 49.44272^2 -(80/2)^2

approx 844.582561

BO approx sqrt(844.582561) approx 29.06170

But BD = 2xxBO -> BD = 58 (0D)