Final answer:
The student is asked to find the ratio of segments AD to DC in a triangle with certain side lengths given. By applying Stewart's Theorem, we find the lengths of AD and DC, leading to the approximate ratio of AD to DC being 0.55.
Step-by-step explanation:
In △ABC, we know that AB=5, BC=7, and AC=9. Furthermore, D is on line segment AC such that BD=5. To find the ratio of AD to DC, we'll use the Stewart's Theorem which is a relation in geometry that relates the lengths of the sides of a triangle to the length of a cevian (a line segment from a vertex to the opposite side). Stewart's Theorem states that man + dad = bmb + cnc for a cevian AD of the triangle, where m and n are the segments into which AD divides the opposite side BC, d is the length of the cevian AD, a, b, and c are the lengths of the sides of the triangle opposite to vertices A, B, and C respectively.
Here m = AD, n = DC and d = BD. Because BD=AB and AB=5, we can replace d with 5. Given AB is equal to BD, and AC is the sum of AD and DC (AD+DC=AC), by substitution we can say that:
AD² + (AD)(DC) = 5² + (DC)(5)
Because AC=9, we replace (AD + DC) with 9.
Let's perform algebraic manipulation to find the value of AD:
- AD² + (AD)(DC) = 25 + 5DC
- AD² + (AD)(9 - AD) = 25 + 5(9 - AD)
- AD² + 9AD - AD² = 45 - 5AD
- 14AD = 45
- AD = 45 / 14
- AD = 3.21 (approx)
Since AC = AD + DC and AC = 9, we can now find DC:
DC = 9 - AD
DC = 9 - 3.21
DC = 5.79 (approx)
Therefore, the ratio of AD to DC is approximately:
A(D)/(D)C = AD / DC
A(D)/(D)C = 3.21 / 5.79
A(D)/(D)C = 0.55 (approx)
The ratio of AD to DC is approximately 0.55.