Exercise 12.1 Q5: Fill in the blanks.

Question: The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle whose area is equal to the sum of the areas of the two circles.

To find the radius of the circle, we need to first find the area of the two given circles and then find the sum of their areas. Let's solve the problem step by step.

Step 1: Find the area of the first circle with radius 19 cm.
The formula to find the area of a circle is given by A = πr², where A is the area and r is the radius.
Substituting the given radius into the formula, we have A1 = π(19)².

Step 2: Find the area of the second circle with radius 9 cm.
Using the same formula, we have A2 = π(9)².

Step 3: Find the sum of the areas of the two circles.
To find the sum of the areas, we add A1 and A2: A = A1 + A2.

Step 4: Find the radius of the circle with the same area as the sum of the areas of the two circles.
Since the area is equal to πr², we can rearrange the formula to find the radius: r = √(A/π).

Substituting the value of A from Step 3 into the formula, we have r = √((A1 + A2)/π).

Step 5: Calculate the radius of the circle.
Now, let's substitute the values of A1 and A2 into the formula and calculate the radius:
r = √((π(19)² + π(9)²)/π)
= √((π(361) + π(81))/π)
= √((442π)/π)
= √442
≈ 21.03 cm

Therefore, the radius of the circle whose area is equal to the sum of the areas of the two given circles is approximately 21.03 cm.