How To Measure The Side Of A Parallelogram Step-by-Step Guide

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Hey guys! Today, we're diving into the fascinating world of parallelograms! We'll tackle a classic geometry problem that many students find tricky. Our goal? To figure out the length of a side in a parallelogram, given some information about its angles and base. Let's get started!

The Problem at Hand

Let's break down the problem we're facing. We need to determine the measure of side 'x' in a parallelogram. We know a couple of key things the opposite angles measure 70° and 110°, and the base of the parallelogram is 10 cm. With this info, we need to pick the correct answer from these options:

  • a) 5 cm
  • b) 10 cm
  • c) 7 cm
  • d) 12 cm

Sounds like a puzzle, right? Don't worry, we'll solve it step by step.

Understanding Parallelograms

Before we jump into calculations, let's make sure we're all on the same page about parallelograms. Think of them as special quadrilaterals (four-sided shapes) with these cool properties:

  • Opposite sides are parallel: This is where the name comes from! Imagine two pairs of lines that never meet, no matter how far you extend them. That's what we mean by parallel.
  • Opposite sides are equal in length: If one side is 10 cm, the side directly across from it is also 10 cm.
  • Opposite angles are equal: Remember those 70° and 110° angles in our problem? They're opposite each other in the parallelogram.
  • Consecutive angles are supplementary: This is a big one! Supplementary angles add up to 180°. So, if one angle is 70°, the angle next to it must be 110° (70° + 110° = 180°).

These properties are our secret weapons for solving parallelogram problems. Let's see how they apply to our specific challenge.

Cracking the Code: Using Angles to Find Sides

Okay, so we know the angles and the base. How do we find the other side? This is where a little trigonometry comes into play. Trigonometry might sound scary, but it's just a fancy way of relating angles and sides in triangles. And guess what? We can create triangles within our parallelogram!

Visualizing the Solution

Imagine drawing a line from one corner of the parallelogram straight down to the base, making a right angle. This line is called the height of the parallelogram. What we've done is split the parallelogram into a rectangle and two right triangles. The right triangles are key because they allow us to use trigonometric ratios like sine, cosine, and tangent.

The Sine Ratio to the Rescue

Remember SOH CAH TOA? This is a handy mnemonic for remembering the trig ratios:

  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

In our case, we're interested in the sine function. Why? Because we know an angle (70° or 110°) and we want to find the side opposite that angle. The side we're trying to find, 'x', will be the hypotenuse of our right triangle.

Setting up the Equation

Let's focus on the triangle formed by the height, the side 'x', and a portion of the base. We'll use the 70° angle for our calculations. Let's call the height 'h'. The sine of 70° is the opposite side (h) divided by the hypotenuse (x):

sin(70°) = h / x

We also need to relate the height to the base. Notice that we've created another right triangle on the other side of the parallelogram. We can use the sine function again, but this time with the 110° angle:

sin(110°) = h / x

Wait a minute! Since sin(110°) = sin(180° - 110°) = sin(70°), we actually have the same equation!

Calculating the Height

Now, we need to find the height 'h'. To do this, we'll use the area of the parallelogram. There are two ways to calculate the area:

  1. Base times height: Area = base * h = 10 cm * h
  2. Using trigonometry: Area = x * 10 cm * sin(70°)

Since both methods calculate the area, we can set them equal to each other:

10 cm * h = x * 10 cm * sin(70°)

Now we can solve for h:

h = x * sin(70°)

Solving for 'x'

We're getting closer! We know that the base of the parallelogram is 10 cm. Let's call the length of the base of our right triangle 'b'. We can use the cosine function to relate 'b' to 'x':

cos(70°) = b / x

So, b = x * cos(70°)

The remaining part of the base will be 10 cm - b.

Now, let's look at the other right triangle. We can use the cosine function with the 110° angle:

cos(110°) = (10 cm - b) / x

But remember, cos(110°) = -cos(70°), so:

-cos(70°) = (10 cm - b) / x

Putting It All Together

Now we have two equations:

  1. b = x * cos(70°)
  2. -cos(70°) = (10 cm - b) / x

Substitute the first equation into the second:

-cos(70°) = (10 cm - x * cos(70°)) / x

Multiply both sides by x:

-x * cos(70°) = 10 cm - x * cos(70°)

This simplifies to:

0 = 10 cm

Oops! This isn't right. It means we need to take a slightly different approach. Let's go back to the basics of parallelograms.

Back to Basics: The Law of Sines (the Easy Way!)

Sometimes, the simplest approach is the best! Instead of diving deep into right triangles and trig ratios, we can use a powerful tool called the Law of Sines. This law is perfect for solving triangles when you know two angles and a side (which we do!).

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. In simpler terms:

a / sin(A) = b / sin(B) = c / sin(C)

Where a, b, and c are the sides of the triangle, and A, B, and C are the opposite angles.

Applying the Law of Sines to Our Parallelogram

Let's focus on one of the triangles we created by drawing a diagonal across the parallelogram. We know one side of this triangle is 10 cm (the base). We also know two angles: 70° and the supplementary angle to 110°, which is also 70°. The third angle in the triangle is 180° - 70° - 70° = 40°.

Now we can use the Law of Sines to find the side 'x'. Let's set up the proportion:

10 cm / sin(110°) = x / sin(40°)

Solving for 'x' (Finally!)

To solve for 'x', we just need to cross-multiply and divide:

x = (10 cm * sin(40°)) / sin(110°)

Now, grab your calculator (make sure it's in degree mode!) and calculate the sines:

sin(40°) ≈ 0.6428 sin(110°) ≈ 0.9397

Plug these values into the equation:

x ≈ (10 cm * 0.6428) / 0.9397 x ≈ 6.84 cm

Choosing the Correct Answer

Looking at our options, the closest answer to 6.84 cm is 7 cm.

The Verdict: The Answer is (c) 7 cm

We did it! We successfully navigated the world of parallelograms, angles, and trig ratios to find the length of side 'x'. Remember, the key is to break down the problem into smaller steps, understand the properties of the shapes you're working with, and choose the right tools (like the Law of Sines) for the job.

Key Takeaways for Parallelogram Problems

  • Know the properties: Opposite sides are parallel and equal, opposite angles are equal, consecutive angles are supplementary.
  • Think triangles: Parallelograms can be divided into triangles, which opens the door to trigonometry.
  • Law of Sines is your friend: This is a powerful tool for solving triangles when you know two angles and a side.
  • Don't be afraid to simplify: Sometimes, the most straightforward approach is the best.
  • Double-check your work: Make sure your calculator is in the correct mode (degrees or radians) and that your answer makes sense in the context of the problem.

Practice Makes Perfect

Geometry can be challenging, but it's also super rewarding. The more you practice, the better you'll become at visualizing shapes, applying formulas, and solving problems. So, keep exploring, keep learning, and keep having fun with math!

If you guys have any questions or want to tackle another geometry problem, let me know in the comments below. Happy problem-solving!