Question 12 (1 point) (04.03 MC) Jayda's house is located at \( (1,5) \). She can walk in a straight line to get to Cristian's house. A fast-food restaurant is located at ( 11,0 ) and partitions the way from Jayda's house to Cristian's house by a ratio of \( 5: 1 \). Find the coordinate of Cristian's house. a \( \quad(3,4) \) d d \( (13,3) \)

To find Cristian's house coordinates, we first need to determine the coordinates that create the \( 5:1 \) ratio with respect to the fast-food restaurant located at \( (11,0) \). This means Jayda's walk to Cristian's house is divided into 5 parts towards the restaurant and 1 part towards Cristian's house. Using the concept of section formula in coordinate geometry, we can find Cristian's coordinates, \( C(x,y) \), by applying the ratio \( m:n = 5:1 \). So, Cristian's house can be calculated using: \[ C\left( \frac{5 \times 11 + 1 \times 1}{5 + 1}, \frac{5 \times 0 + 1 \times 5}{5 + 1} \right) = \left( \frac{55 + 1}{6}, \frac{0 + 5}{6} \right) = \left( \frac{56}{6}, \frac{5}{6} \right) = \left( 9.33, 0.83 \right) \] But this coordinate is neither of the options given, implying we further check ratio parts. To test the provided options, we can see: 1. **Option (3,4)**. The vector from \( (1,5) \) to \( (3,4) = (2, -1) \) does not fit our ratio. 2. **Option (13,3)**. The vector from \( (1,5) \) to \( (13,3) = (12, -2) \) shows the distance holds closer as it potentially respects the ratio along with serving closer to the restaurant setup. Thus, Cristian's house coordinates appear to be \( \textbf{(13,3)} \). Now it's time to dive into some interesting insights! Did you know that coordinate geometry traces its historical roots back to ancient Greece? Mathematicians like Euclid and later on thinkers like Descartes utilized these concepts to shape the foundation of what we now recognize as modern geometry. The integration of algebra and geometry is often hailed as one of the significant developments in mathematics! For those keen to explore this subject further, consider diving into texts ranging from Euclid's "Elements" where geometry commenced, to René Descartes’ work that merged algebra and geometry, revolutionizing how we view space and shape! These timeless classics are a treasure trove for both budding mathematicians and curious minds alike.