ABCD is a rectangle with sides AB = 56 cm and BC = 45 cm, and E is the midpoint of side CD. Then, the length, in cm, of radius of incircle of △ADE is

Question

Accepted Answer

The problem involves finding the inradius of the right-angled triangle $$ 	riangle ADE $$, which is formed within a rectangle. Here's the step-by-step solution:

**Given Dimensions:**

- Rectangle $$ ABCD $$:
     - $$ AB = 56 \, 	ext{cm} $$
     - $$ BC = 45 \, 	ext{cm} $$
   - Point $$ E $$ is the midpoint of side $$ CD $$, so:
     $$
     DE = \frac{CD}{2} = \frac{AB}{2} = 28 \, 	ext{cm}.
     $$
   - Side $$ AD = BC = 45 \, 	ext{cm}. $$

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**Apply Pythagoras Theorem:**

- In right-angled triangle $$ 	riangle ADE $$, where $$ AD $$ and $$ DE $$ are the legs and $$ AE $$ is the hypotenuse:
     $$
     AE^2 = AD^2 + DE^2
     $$
   - Substituting values:
     $$
     AE^2 = 45^2 + 28^2 = 2025 + 784 = 2809
     $$
   - Solve for $$ AE $$:
     $$
     AE = \sqrt{2809} = 53 \, 	ext{cm}.
     $$

**Formula for Inradius of a Right-Angled Triangle:**

- The inradius ($$ r $$) of a right-angled triangle is given by:
     $$
     r = \frac{a + b - h}{2}
     $$
     where $$ a $$ and $$ b $$ are the legs, and $$ h $$ is the hypotenuse.

**Calculate Inradius:**

- Substituting values:
     $$
     r = \frac{45 + 28 - 53}{2}
     $$
   - Simplify:
     $$
     r = \frac{20}{2} = 10 \, 	ext{cm}.
     $$

### Final Answer:
The length of the radius of the incircle of $$ 	riangle ADE $$ is **10 cm**.  
Correct option: **10**

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