Answer :
To find the measure of the obtuse angle in a rhombus with given side lengths and one diagonal length, we can utilize some geometric properties and trigonometric rules. Here's a step-by-step solution:
1. Define Properties of the Rhombus:
- All sides are equal.
- The diagonals bisect each other at right angles.
- Each diagonal splits the rhombus into two congruent triangles.
2. Given Information:
- Side length ([tex]\( a \)[/tex]) = 50 yards
- One of the diagonals ([tex]\( d_2 \)[/tex]) = 95 yards
3. Divide the Problem Using Geometry:
- Since the diagonals bisect each other at right angles, let's denote [tex]\( d_1 \)[/tex] as the length of the other diagonal.
- Each half of [tex]\( d_2 \)[/tex] is [tex]\( d_2/2 = 95/2 = 47.5 \)[/tex] yards.
4. Form a Triangle:
- Let's consider the right triangle formed by half the diagonal [tex]\( d_2 \)[/tex], half the diagonal [tex]\( d_1 \)[/tex], and the side length of the rhombus.
- Using the Pythagorean theorem in the right triangle:
[tex]\[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2 \][/tex]
Substituting the values we know:
[tex]\[ \left(\frac{d_1}{2}\right)^2 + (47.5)^2 = (50)^2 \][/tex]
[tex]\[ \left(\frac{d_1}{2}\right)^2 + 2256.25 = 2500 \][/tex]
[tex]\[ \left(\frac{d_1}{2}\right)^2 = 2500 - 2256.25 \][/tex]
[tex]\[ \left(\frac{d_1}{2}\right)^2 = 243.75 \][/tex]
[tex]\[ \frac{d_1}{2} = \sqrt{243.75} \][/tex]
[tex]\[ \frac{d_1}{2} \approx 15.61 \text{ yards} \][/tex]
Therefore:
[tex]\[ d_1 \approx 2 \times 15.61 = 31.22 \text{ yards} \][/tex]
5. Use Cosine Rule to Find the Obtuse Angle:
- In a rhombus, one pair of opposite angles are obtuse and the other pair are acute. We need the obtuse angle.
- Considering the triangle with sides [tex]\( a, a, d_2 \)[/tex], and using the cosine rule:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Here, [tex]\( c = d_2 \)[/tex], and [tex]\( a = b = 50 \)[/tex], and we need to find [tex]\( \cos(C) \)[/tex]:
[tex]\[ (95)^2 = 50^2 + 50^2 - 2 \times 50 \times 50 \times \cos(C) \][/tex]
[tex]\[ 9025 = 2500 + 2500 - 5000 \cos(C) \][/tex]
[tex]\[ 9025 = 5000 - 5000 \cos(C) \][/tex]
[tex]\[ 5000 \cos(C) = 5000 - 9025 \][/tex]
[tex]\[ 5000 \cos(C) = -4025 \][/tex]
[tex]\[ \cos(C) = -0.805 \][/tex]
[tex]\[ C = \arccos(-0.805) \][/tex]
[tex]\[ C \approx 143.5^\circ \][/tex]
6. Result:
- The obtuse angle of the rhombus is approximately 144 degrees to the nearest whole degree.
So, the measure of the obtuse angle in the rhombus to the nearest whole degree is 144 degrees.
1. Define Properties of the Rhombus:
- All sides are equal.
- The diagonals bisect each other at right angles.
- Each diagonal splits the rhombus into two congruent triangles.
2. Given Information:
- Side length ([tex]\( a \)[/tex]) = 50 yards
- One of the diagonals ([tex]\( d_2 \)[/tex]) = 95 yards
3. Divide the Problem Using Geometry:
- Since the diagonals bisect each other at right angles, let's denote [tex]\( d_1 \)[/tex] as the length of the other diagonal.
- Each half of [tex]\( d_2 \)[/tex] is [tex]\( d_2/2 = 95/2 = 47.5 \)[/tex] yards.
4. Form a Triangle:
- Let's consider the right triangle formed by half the diagonal [tex]\( d_2 \)[/tex], half the diagonal [tex]\( d_1 \)[/tex], and the side length of the rhombus.
- Using the Pythagorean theorem in the right triangle:
[tex]\[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2 \][/tex]
Substituting the values we know:
[tex]\[ \left(\frac{d_1}{2}\right)^2 + (47.5)^2 = (50)^2 \][/tex]
[tex]\[ \left(\frac{d_1}{2}\right)^2 + 2256.25 = 2500 \][/tex]
[tex]\[ \left(\frac{d_1}{2}\right)^2 = 2500 - 2256.25 \][/tex]
[tex]\[ \left(\frac{d_1}{2}\right)^2 = 243.75 \][/tex]
[tex]\[ \frac{d_1}{2} = \sqrt{243.75} \][/tex]
[tex]\[ \frac{d_1}{2} \approx 15.61 \text{ yards} \][/tex]
Therefore:
[tex]\[ d_1 \approx 2 \times 15.61 = 31.22 \text{ yards} \][/tex]
5. Use Cosine Rule to Find the Obtuse Angle:
- In a rhombus, one pair of opposite angles are obtuse and the other pair are acute. We need the obtuse angle.
- Considering the triangle with sides [tex]\( a, a, d_2 \)[/tex], and using the cosine rule:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Here, [tex]\( c = d_2 \)[/tex], and [tex]\( a = b = 50 \)[/tex], and we need to find [tex]\( \cos(C) \)[/tex]:
[tex]\[ (95)^2 = 50^2 + 50^2 - 2 \times 50 \times 50 \times \cos(C) \][/tex]
[tex]\[ 9025 = 2500 + 2500 - 5000 \cos(C) \][/tex]
[tex]\[ 9025 = 5000 - 5000 \cos(C) \][/tex]
[tex]\[ 5000 \cos(C) = 5000 - 9025 \][/tex]
[tex]\[ 5000 \cos(C) = -4025 \][/tex]
[tex]\[ \cos(C) = -0.805 \][/tex]
[tex]\[ C = \arccos(-0.805) \][/tex]
[tex]\[ C \approx 143.5^\circ \][/tex]
6. Result:
- The obtuse angle of the rhombus is approximately 144 degrees to the nearest whole degree.
So, the measure of the obtuse angle in the rhombus to the nearest whole degree is 144 degrees.
