Question

A rectangle \(ABCD\) has sides \(AB = 45 \, \text{cm}\) and \(BC = 26 \, \text{cm}\). Point \(E\) is the midpoint of side \(CD\). Find the radius of the incircle of the triangle \(\triangle AED\).

Answer 1

Correct Answer: 7.07

Rectangle $ABCD$ with sides:
\[AB = 45 \, \text{cm}, \quad BC = 26 \, \text{cm}.\]
$E$ is the midpoint of $CD$, so:
\[CE = ED = \frac{CD}{2} = \frac{45}{2} = 22.5 \, \text{cm}.\]
Coordinates of points:
\[A(0, 0), \, B(45, 0), \, D(0, 26), \, C(45, 26), \, E(22.5, 26).\]
Step 1: Calculate the lengths of the sides of $\triangle AED$
1. Length of $AE$:
\[AE = \sqrt{(22.5 - 0)^2 + (26 - 0)^2} = \sqrt{22.5^2 + 26^2}.\]
Simplifying:
\[AE = \sqrt{506.25 + 676} = \sqrt{1182.25} \approx 34.39 \, \text{cm}.\]
2. Length of $ED$:
\[ED = 22.5 \, \text{cm}.\]
3. Length of $AD$:
\[AD = 26 \, \text{cm}.\]
Step 2: Calculate the area of $\triangle AED$
Using Heron's formula, the semi-perimeter ($s$) is:
\[s = \frac{AE + ED + AD}{2} = \frac{34.39 + 22.5 + 26}{2} = 41.445 \, \text{cm}.\]
The area ($\Delta$) is given by:
\[\Delta = \sqrt{s(s - AE)(s - ED)(s - AD)}.\]
Substitute the values:
\[\Delta = \sqrt{41.445 \cdot (41.445 - 34.39) \cdot (41.445 - 22.5) \cdot (41.445 - 26)}.\]
Simplify each term:
\[\Delta = \sqrt{41.445 \cdot 7.055 \cdot 18.945 \cdot 15.445}.\]
\[\Delta \approx \sqrt{85952.84} \approx 293.17 \, \text{cm}^2.\]
Step 3: Radius of the incircle  
The radius of the incircle ($r$) is given by:
\[r = \frac{\Delta}{s}.\]
Substitute the values:
\[r = \frac{293.17}{41.445} \approx 7.07 \, \text{cm}.\]