Recognizing common Pythagorean triples like (3, 4, 5) can save time. When you see a right triangle with legs 3 and 4, you should immediately know the hypotenuse is 5.
Step 1: Understanding the Concept:
The problem involves finding a missing side in a right-angled triangle inscribed in a circle. The key concept is that the hypotenuse of both triangles is the radius of the circle, and we can use the Pythagorean theorem. Step 2: Key Formula or Approach:
The Pythagorean theorem states that for a right-angled triangle with legs \(a\) and \(b\) and hypotenuse \(c\), we have \(a^2 + b^2 = c^2\). Step 3: Detailed Explanation: Find the radius of the circle:
Look at the bottom right-angled triangle. Its legs are of length 3 and 4. The hypotenuse is the line segment from the center O to the circle's edge, which is the radius (let's call it \(r\)).
Using the Pythagorean theorem:
\[ r^2 = 3^2 + 4^2 \]
\[ r^2 = 9 + 16 \]
\[ r^2 = 25 \]
\[ r = \sqrt{25} = 5 \]
So, the radius of the circle is 5. Find the value of x:
Now look at the other right-angled triangle. Its legs are of length 3 and \(x\). Its hypotenuse is also the radius of the circle, which we found to be 5.
Using the Pythagorean theorem again:
\[ 5^2 = x^2 + 3^2 \]
\[ 25 = x^2 + 9 \]
Subtract 9 from both sides:
\[ 25 - 9 = x^2 \]
\[ 16 = x^2 \]
\[ x = \sqrt{16} = 4 \]
Since \(x\) represents a length, we take the positive root. So, \(x=4\). Step 4: Final Answer:
We are comparing Column A (\(x\)) and Column B (5).
Column A = 4
Column B = 5
Since \(4<5\), the quantity in Column B is greater.
