If the area of a circle is equal to the sum of the areas of two circles with diameters of 10 cm and 24 cm, then find the diameter of the larger circle (in cm).

Given:

The diameter of the first circle = $10 \mathrm{cm}$

Since the diameter is always twice the radius.

So, radius, r₁ = $\frac{\mathbf{Diameter}}{\mathbf{2}}$

                       = $\frac{10}{2}$ cm = $\mathbf{5}\mathbf{ }\mathbf{cm}$

And diameter of the second circle = $24 \mathrm{cm}$

So, radius, r₂ = $\frac{\mathbf{Diameter}}{\mathbf{2}}$

                       = $\frac{24}{2}$ cm = $\mathbf{12}\mathbf{ }\mathbf{cm}$

Now, we know the area of a circle is ${\mathbf{\pi R}}^{\mathbf{2}}$.

So, the area of the first circle = ${{\mathbf{\pi r}}_{\mathbf{1}}}^{\mathbf{2}}$

And, the area of the second circle = ${{\mathbf{\pi r}}_{\mathbf{2}}}^{\mathbf{2}}$ 

The sum of the areas = ${{\mathrm{\pi r}}_1}^2 + {{\mathrm{\pi r}}_2}^2$

                                      = ${\mathbf{\pi }\mathbf{(}{\mathbf{r}}_{\mathbf{1}}}^{\mathbf{2}} + {{\mathbf{\pi r}}_{\mathbf{2}}}^{\mathbf{2}})$

Substituting the values, we get:

= $\mathrm{\pi }(5^2 + {12}^2)$ 

= $\mathrm{\pi }\times (25 + 144)$ 

= $\mathbf{169}\mathbf{\pi }\mathbf{ }{\mathbf{cm}}^{\mathbf{2}}$ 

According to the question, the sum of the areas of the two circles is the area of the larger circle.

So, the area of the large circle = $\mathbf{169}\mathbf{\pi }$ cm²

Let its radius be $\mathbf{R}\mathbf{ }\mathbf{cm}\mathbf{.}$

$\Rightarrow {\mathrm{\pi R}}^2 = 169\mathrm{\pi }$

$\Rightarrow {\mathrm{R}}^2 = 169$

$\Rightarrow \mathrm{R} = \sqrt{169 }$

$\Rightarrow \mathbf{R}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{13}\mathbf{ }\mathbf{cm}$

Thus, the diameter = $\mathbf{2}\mathbf{\times }\mathbf{R}$

                                  = $2\times 13$ cm = $\mathbf{26}\mathbf{ }\mathbf{cm}$

Hence, the diameter will be $\mathbf{26}\mathbf{ }\mathbf{cm}$.