Math Problem Statement

a certain city block is in the form of parallelogram two of its sides are each 35m long and the other side is 22m each if the distance between the first pair is 12 find the length of the diagonal

Solution

To find the length of the diagonal of a parallelogram, we can use the formula derived from the cosine rule. For a parallelogram with sides aa and bb, and an angle θ\theta between them, the lengths of the diagonals d1d_1 and d2d_2 can be calculated as follows:

d1=a2+b2+2abcos(θ)d_1 = \sqrt{a^2 + b^2 + 2ab \cos(\theta)} d2=a2+b22abcos(θ)d_2 = \sqrt{a^2 + b^2 - 2ab \cos(\theta)}

In your case:

  • a=35ma = 35 \, m
  • b=22mb = 22 \, m
  • The distance between the two sides (height) is given as 12 m, but we need the angle θ\theta or the diagonal directly.

Instead, we can find the length of the diagonal using the relationship between the sides and the height. Since the height forms a right triangle with the sides of the parallelogram, we can use the Pythagorean theorem.

The formula for the diagonals can also be simplified for our situation. The diagonals of a parallelogram can be expressed as:

d=a2+b22abcos(θ)d = \sqrt{a^2 + b^2 - 2ab \cos(\theta)}

Given that we don’t have the angle directly, we can compute using the following steps.

  1. Use the height to determine the angle between the sides.

The height hh and side bb (22 m) can help us find the angle:

sin(θ)=hb    h=bsin(θ)    12=22sin(θ)\sin(\theta) = \frac{h}{b} \implies h = b \sin(\theta) \implies 12 = 22 \sin(\theta)

sin(θ)=1222=611\sin(\theta) = \frac{12}{22} = \frac{6}{11}

Now, we can find cos(θ)\cos(\theta) using sin2(θ)+cos2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1:

cos2(θ)=1(611)2=136121=85121\cos^2(\theta) = 1 - \left(\frac{6}{11}\right)^2 = 1 - \frac{36}{121} = \frac{85}{121} cos(θ)=85121=8511\cos(\theta) = \sqrt{\frac{85}{121}} = \frac{\sqrt{85}}{11}

  1. Substituting into the diagonal formula.

Now we can substitute back into the diagonal formula for one of the diagonals d1d_1:

d=352+222235228511d = \sqrt{35^2 + 22^2 - 2 \cdot 35 \cdot 22 \cdot \frac{\sqrt{85}}{11}}

Calculating this step by step:

d=1225+484235228511d = \sqrt{1225 + 484 - 2 \cdot 35 \cdot 22 \cdot \frac{\sqrt{85}}{11}} d=170915408511d = \sqrt{1709 - \frac{1540\sqrt{85}}{11}}

To simplify this further, let's calculate numerically:

352=1225,222=484,23522=154035^2 = 1225, \quad 22^2 = 484, \quad 2 \cdot 35 \cdot 22 = 1540

Thus, we compute:

d1709140.68 (where 859.22)d \approx \sqrt{1709 - 140.68} \text{ (where } \sqrt{85} \approx 9.22) d1568.3239.6 md \approx \sqrt{1568.32} \approx 39.6 \text{ m}

Result

The length of the diagonal is approximately 39.6m39.6 \, m.

Would you like more details on any of the calculations or concepts? Here are some related questions to consider:

  1. How do you calculate the area of a parallelogram?
  2. What properties do the diagonals of a parallelogram have?
  3. How does the height relate to the angles in a parallelogram?
  4. Can you derive the formula for the diagonal without using trigonometric functions?
  5. What is the significance of the sides being equal in a parallelogram?

Tip: Always check if you have sufficient information before proceeding with formulas, especially in geometry!

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Math Problem Analysis

Mathematical Concepts

Geometry
Parallelograms
Trigonometry

Formulas

d_1 = √(a² + b² + 2ab cos(θ))
d_2 = √(a² + b² - 2ab cos(θ))
h = b sin(θ)

Theorems

Pythagorean Theorem
Cosine Rule

Suitable Grade Level

Grades 9-12