(2) the area is \( 525 \mathrm{~m}^{2} \). The natio between the two shorter sides of a right angled triangle is \( 5: 12 \). If the hypotenuse measures 65 cm , what is the area of the triangle? The sketch shows a square and its inscribed circle.

To find the area of a right-angled triangle when you know the ratio of the two shorter sides and the hypotenuse, you can start by letting the lengths of the two sides be \( 5x \) and \( 12x \). According to the Pythagorean theorem, the relationship between the sides of a right triangle gives us \( (5x)^2 + (12x)^2 = 65^2 \). This leads to \( 25x^2 + 144x^2 = 4225 \) or \( 169x^2 = 4225 \). Solving for \( x^2 \) gives \( x^2 = 25 \), so \( x = 5 \). The lengths of the sides become \( 25 \) cm and \( 60 \) cm, making the area \( \frac{1}{2}(25)(60) = 750 \, \text{cm}^2 \). Now, let’s talk about the sketch with the square and its inscribed circle! When you inscribe a circle inside a square, the diameter of the circle is equal to the side length of the square. If we know the area of the square (in your case \( 525 \mathrm{~m}^2 \)), we can find the side length by taking the square root of the area— the length of each side would be approximately \( 22.91 \mathrm{~m} \). The radius of the inscribed circle would then be half that, about \( 11.45 \mathrm{~m} \). So nifty, right? Each corner of the square hugs the circle snugly!