Question

△cba ~ △clk
find the length of \\(\overline{al}\\):
answer

Explanation:

Step1: Identify Similar Triangles Property

Since $\triangle CBA \sim \triangle CLK$, their corresponding sides are proportional. Let's assume $AC = x$, then $AL = AC + CL = x + 14$. Wait, actually, looking at the segments, $BC = 56$ and $XC = 8$ (wait, the diagram has $XC = 8$? Wait, the user's diagram: $BC = 56$, $CL = 14$, $XC = 8$? Wait, maybe the sides: in similar triangles, $\frac{BC}{XC}=\frac{AC}{LC}$. Wait, let's correct: $\triangle CBA \sim \triangle CLK$, so $\frac{BC}{KC}=\frac{AC}{LC}=\frac{BA}{LK}$. Wait, maybe the segments: $BC = 56$, $KC = 8$ (wait, the diagram has $XC$ as 8? Wait, the original problem: the triangle $\triangle CBA \sim \triangle CLK$, with $BC = 56$, $LC = 14$, $KC = 8$? Wait, maybe the correct proportion is $\frac{BC}{KC}=\frac{AC}{LC}$. Wait, let's re-express: Let $AC = y$, then $AL = AC + CL = y + 14$. From similarity, $\frac{BC}{KC}=\frac{AC}{LC}$. So $\frac{56}{8}=\frac{y}{14}$. Wait, solving for $y$: $y=\frac{56\times14}{8}= 98$? No, wait, maybe I mixed up the segments. Wait, maybe $BC = 56$, $XC = 8$ (wait, the diagram has $X$ and $L$? Wait, maybe the triangles are $\triangle CBA$ and $\triangle CXL$? Wait, the user's diagram: $\triangle CBA \sim \triangle CLK$ (maybe typo, $\triangle CXL$). Let's assume the correct proportion: $\frac{BC}{XC}=\frac{AC}{LC}$. So $BC = 56$, $XC = 8$, $LC = 14$. Then $\frac{56}{8}=\frac{AC}{14}$. So $AC=\frac{56\times14}{8}= 98$? No, that can't be. Wait, maybe the other way: $\frac{BC}{LC}=\frac{AC}{XC}$. So $BC = 56$, $LC = 14$, $XC = 8$. Then $\frac{56}{14}=\frac{AC}{8}$. So $AC = \frac{56\times8}{14}= 32$? Wait, no, 56 divided by 14 is 4, so $AC = 4\times8 = 32$. Then $AL = AC + CL = 32 + 14 = 46$? No, that's not right. Wait, maybe the segments are $BC = 56$, $XC = 8$, $CL = 14$, and $AC$ is the other side. Wait, maybe the correct proportion is $\frac{BC}{XC}=\frac{AC}{LC}$. So $56/8 = AC/14$. Then $AC = (56\times14)/8 = 98$. Then $AL = AC + CL = 98 + 14 = 112$? No, that's too big. Wait, maybe I misread the diagram. Wait, the problem says "Find the length of $\overline{AL}$". Let's re-express: Let's assume that $BC = 56$, $XC = 8$, $LC = 14$, and $\triangle CBA \sim \triangle CXL$ (maybe the similar triangle is $\triangle CXL$). So corresponding sides: $BC$ corresponds to $XC$, $AC$ corresponds to $LC$, $BA$ corresponds to $XL$. So $\frac{BC}{XC}=\frac{AC}{LC}$. So $56/8 = AC/14$. Then $AC = (56\times14)/8 = 98$? No, 56 divided by 8 is 7, so 7 times 14 is 98. Then $AL = AC + CL = 98 + 14 = 112$? Wait, that seems off. Wait, maybe the similar triangles have $BC = 56$, $KC = 8$ (wait, the diagram has $XC$ as 8, $LC$ as 14). Wait, maybe the correct proportion is $\frac{BC}{LC}=\frac{AC}{XC}$. So $56/14 = AC/8$. Then $AC = (56/14)\times8 = 4\times8 = 32$. Then $AL = AC + CL = 32 + 14 = 46$? No, that's not. Wait, maybe the segments are $BC = 56$, $XC = 8$, $AC = 14$, and we need to find $AL$. No, the diagram shows $CL = 14$, $XC = 8$, $BC = 56$. Wait, maybe the similar triangles: $\triangle CBA \sim \triangle CXL$, so $\frac{BC}{XC}=\frac{AC}{XL}$? No, I think I made a mistake. Wait, let's start over.

Given $\triangle CBA \sim \triangle CLK$ (or $\triangle CXL$), so the ratio of corresponding sides is equal. Let's denote:

- $BC = 56$
- $KC = 8$ (or $XC = 8$)
- $LC = 14$
- $AC =?$
- $AL = AC + LC = AC + 14$

From similarity, $\frac{BC}{KC}=\frac{AC}{LC}$

So $\frac{56}{8}=\frac{AC}{14}$

Step2: Solve for AC

Cross-multiplying: $AC = \frac{56 \times 14}{8}$

Simplify: $56 \div 8 = 7$, so $AC = 7 \times 14 = 98$

Step3: Find AL

$AL = AC + LC = 98 + 14 = 112$? Wait, no, that can't be. Wait, maybe $LC$ is $AC$? No, the diagram shows $A$ and $L$ on a line, with $C$ between them. So $AC$ and $CL$ are segments of $AL$, so $AL = AC + CL$. Wait, but if $AC = 98$ and $CL = 14$, then $AL = 98 + 14 = 112$. But that seems large. Wait, maybe the ratio is reversed. Maybe $\frac{KC}{BC}=\frac{LC}{AC}$. So $\frac{8}{56}=\frac{14}{AC}$. Then $AC = \frac{56 \times 14}{8}= 98$, same as before. So $AL = 98 + 14 = 112$.

Wait, maybe the correct answer is 112. But let's check again.

Wait, maybe the similar triangles have $BC = 56$, $XC = 8$, $LC = 14$, and $AC$ is the side from $C$ to $A$, so $AL = AC + CL$. So with the proportion $\frac{BC}{XC}=\frac{AC}{LC}$, we get $AC = (56 \times 14)/8 = 98$, then $AL = 98 + 14 = 112$.

Answer:

112