Video: CBSE Class X • Pack 4 • 2015 • Question 21

CBSE Class X • Pack 4 • 2015 • Question 21

05:20

Video Transcript

The diagonal of a rectangular field is 16 metres longer than its shorter side. If the longer side is 14 metres longer than the shorter side, find the lengths of the sides of the field.

We should begin by sketching a diagram of the field. Remember it doesn’t need to be to scale. But it should be roughly in proportion so we can identify how to solve this problem. It’s a rectangular field. So we know each of its interior angles are 90 degrees.

We can also call the shortest side of the field 𝑥 metres. The diagonal is 16 metres longer. Let’s call that 𝑥 plus 16 metres. Since the longer side is 14 metres longer than the shorter side, we can call that 𝑥 plus 14 metres. Notice that by including the diagonal of the rectangular field, we’ve created two identical right-angled triangles. This means we can use the Pythagorean theorem to help us calculate the value of 𝑥.

Remember the Pythagorean theorem says that the square of the hypotenuse — that’s the longest side — is equal to the sum of the squares of the two smaller sides. It’s often written as 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the length of the hypotenuse.

Let’s substitute what we know about our right-angled triangle into this formula. The diagonal is the hypotenuse; that’s 𝑥 plus 16. So our formula becomes 𝑥 squared plus 𝑥 plus 14 squared equals 𝑥 plus 16 squared.

Our next step is to expand the brackets before we do anything else. A common mistake here is to think that to square a bracket, we just square its two terms. In fact, we should write it as the product of two brackets and then expand fully.

We first multiply the first term in each bracket. 𝑥 multiplied by 𝑥 is 𝑥 squared. We then multiply the outer terms. That’s 𝑥 multiplied by 14, which is 14𝑥. Multiplying the inner terms — that’s 14 multiplied by 𝑥 — is once again 14𝑥. And then, 14 multiplied by 14 is 196. It’s also sensible to simplify by collecting like terms. That gives us 𝑥 squared plus 28𝑥 plus 196.

We can repeat this process for 𝑥 plus 16 squared. 𝑥 multiplied by 𝑥 is 𝑥 squared. 𝑥 multiplied by 16 is 16𝑥. 16 multiplied by 𝑥 gives us another 16𝑥. And 16 multiplied by 16 is 256. This simplifies to 𝑥 squared plus 32𝑥 plus 256. Substituting these back into our original equation and we get 𝑥 squared plus 𝑥 squared plus 28𝑥 plus 196 is equal to 𝑥 squared plus 32𝑥 plus 256.

We have a quadratic equation. And to solve this, we first need to rearrange so that we have a quadratic expression that’s equal to zero. Let’s subtract 𝑥 squared from both sides. Doing so leaves us with 𝑥 squared plus 28𝑥 plus 196 is equal to 32𝑥 plus 256. Next, we’ll subtract 32𝑥 from both sides. And that leaves us with 𝑥 squared minus four 𝑥 plus 196 equals 256. Finally, we’ll subtract 256 from both sides. That gives us 𝑥 squared minus four 𝑥 minus 60 is equal to zero.

Now, we need to factorize the expression on the left-hand side. We know that it will consist of two brackets. And in the front of each bracket, there will be an 𝑥. We know this because when we multiply these brackets back out again, 𝑥 multiplied by 𝑥 gives us this 𝑥 squared that’s required.

To find the constant in each bracket, we need to find the factor pair of negative 60 that has a sum of negative four. Negative 60 has quite a few factor pairs. Let’s narrow it down to 10 and negative six or six and negative 10. We’ve done this because 10 and six have a difference of four. In fact, the factor pairs that sums to make negative four is six and negative 10.

So factorizing our expression, we get 𝑥 plus six multiplied by 𝑥 minus 10 is equal to zero. Now, we have two brackets whose product is zero. For this to be true, either bracket must individually be equal to zero: so 𝑥 plus six is equal to zero or 𝑥 minus 10 is equal to zero.

To solve this first equation, we’ll subtract six from both sides. And that gives us 𝑥 is equal to negative six. To solve the second equation, we’ll add 10 to both sides. That gives us that 𝑥 is equal to 10. In fact, we said that 𝑥 was the length of the shorter side of the field. 𝑥 cannot, therefore, be negative.

That means that the shorter side of the field must be 10 metres in length. The longer side of the field is 14 metres longer. 10 plus 14 is 24. So the longer side of the field is 24 metres.

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