In the figure given below, triangle ABC has a right angle mark at vertex B, with BD drawn from B to the base AC meeting it at D with a right angle there. AB = 4x, BC = 5y+10, AD = 2x, and DC = 3y. The value of x and y would be ________
Show Hint
Use the two smaller right triangles formed by the altitude to write two equations in x and y.
Step 1: Identify the two smaller right triangles inside the big triangle.
The line from B down to the base meets AC at a point, call it D, at a right angle. This splits the big triangle into triangle ABD (left) and triangle CBD (right), and both share the side BD.
Step 2: Use AAS to show the two smaller triangles are congruent.
Both small triangles have a right angle at D, both share side BD, and (from the figure) the angles at B on either side of BD are equal, so by the AAS rule, triangle ABD is congruent to triangle CBD. Matching sides then give $AD = DC$ and $AB = BC$.
Step 3: Write the two linear equations and eliminate y.
$AD=DC$ gives $2x = 3y$, that is $2x - 3y = 0$. $AB = BC$ gives $4x = 5y+10$, that is $4x - 5y = 10$. Multiply the first equation by 2 to line up the x terms: $4x - 6y = 0$. Subtract this from the second equation: $(4x-5y) - (4x-6y) = 10 - 0$, which simplifies to $y = 10$.
Step 4: Back substitute to get x.
Put $y=10$ into $2x - 3y = 0$: $2x = 30$, so $x = 15$. This matches the value found by direct substitution.
Final Answer:
$x = 15$ and $y = 10$.
\[ \boxed{x=15,\ y=10} \]
