Question

express x in terms of a, b, and c

Explanation:

Step1: Identify Similar Triangles

The two triangles are similar (AA similarity, same angle \( A \) and common angle at the vertex). For similar triangles, corresponding sides are proportional. So, \(\frac{x}{x + a}=\frac{c}{b}\) (wait, no, correct proportion: smaller triangle sides \( x, c \); larger triangle sides \( x + a, b \)? Wait, no, looking at the diagram: the top triangle has sides \( x \) (one leg) and \( c \) (the other leg), and the bottom triangle (combined with top) has leg \( x + a \) and \( b \)? Wait, no, maybe the parallel line \( b \) and \( c \)? Wait, actually, the two triangles: the smaller one (lower) has sides \( a \) and... Wait, no, let's re-express. Let the two triangles be similar, so the ratio of corresponding sides is equal. Let the top triangle have sides \( x \) (adjacent to angle \( A \)) and \( c \) (opposite? Wait, no, angle \( A \) is common, and the line \( b \) is parallel to \( c \), so the triangles are similar by AA (angle \( A \) and the right angle? Wait, maybe the triangles are similar, so \(\frac{x}{a}=\frac{c}{b}\)? No, wait, let's set up the proportion correctly. If the two triangles are similar, then \(\frac{x}{x + a}=\frac{c}{b}\)? No, that doesn't make sense. Wait, maybe the top triangle has sides \( x \) (one side) and \( c \) (another), and the bottom triangle (the smaller one) has sides \( a \) (corresponding to \( x \)) and... Wait, no, the correct proportion for similar triangles: if triangle 1 has sides \( x \) and \( c \), triangle 2 has sides \( a \) and \( b - c \)? No, the diagram shows a line segment \( b \) parallel to \( c \), creating two similar triangles. So the larger triangle has sides \( x + a \) (the leg) and \( b \) (the base), and the smaller triangle (top) has sides \( x \) (leg) and \( c \) (base). So by similarity, \(\frac{x}{x + a}=\frac{c}{b}\). Now solve for \( x \).

Step2: Cross - Multiply

Cross - multiplying the proportion \(\frac{x}{x + a}=\frac{c}{b}\) gives \( b\times x=c\times(x + a)\).

Step3: Expand and Solve for \( x \)

Expand the right - hand side: \(bx=cx+ca\).
Subtract \(cx\) from both sides: \(bx - cx=ca\).
Factor out \(x\) on the left - hand side: \(x(b - c)=ca\).
Then, solve for \(x\): \(x=\frac{ac}{b - c}\)? Wait, no, wait, maybe I mixed up the sides. Wait, maybe the correct proportion is \(\frac{x}{a}=\frac{c}{b - c}\)? No, let's re - examine. Wait, the two triangles: the top triangle has base \( c \) and leg \( x \), the bottom triangle (the smaller one) has base \( b - c \) and leg \( a \)? No, the diagram shows a line \( b \) (the longer base) and \( c \) (the shorter base), with the leg of the top triangle being \( x \) and the leg of the bottom triangle being \( a \). So the two triangles are similar, so \(\frac{x}{x + a}=\frac{c}{b}\) is wrong. Wait, actually, the correct proportion is \(\frac{x}{a}=\frac{c}{b - c}\)? No, let's do it again. Let the big triangle have legs \( x + a \) (vertical) and \( b \) (horizontal), and the small triangle (top) have legs \( x \) (vertical) and \( c \) (horizontal). Since they are similar, \(\frac{x}{x + a}=\frac{c}{b}\). Cross - multiply: \(bx=cx+ca\). Then \(bx - cx=ca\), \(x(b - c)=ca\), so \(x=\frac{ac}{b - c}\)? Wait, no, that would be if \(b>c\). Wait, maybe the other way: \(\frac{x}{a}=\frac{c}{b}\), no, that's not. Wait, maybe the triangles are similar with ratio \(\frac{x}{a}=\frac{c}{b - c}\)? No, I think I made a mistake in the proportion. Wait, the correct approach: if two triangles are similar, the ratio of corresponding sides is equal. Let the top triangle have sides \( x \) (adjacent to angle \( A \)) and \( c \) (opposite), and the bottom triangle (the one with side \( a \)) has sides \( a \) (adjacent) and \( b - c \) (opposite). Then \(\frac{x}{a}=\frac{c}{b - c}\), so \(x=\frac{ac}{b - c}\)? No, that doesn't seem right. Wait, maybe the line \( b \) is parallel to \( c \), so the triangles are similar, so \(\frac{x}{x + a}=\frac{c}{b}\). Let's solve this equation:

\(bx = c(x + a)\)

\(bx=cx+ca\)

\(bx - cx=ca\)

\(x(b - c)=ca\)

\(x=\frac{ac}{b - c}\)

Wait, but maybe the correct proportion is \(\frac{x}{a}=\frac{c}{b}\), which would give \(x=\frac{ac}{b}\), but that would be if the triangles are not nested. Wait, the diagram shows a nested triangle, so the correct proportion is \(\frac{x}{x + a}=\frac{c}{b}\), leading to \(x=\frac{ac}{b - c}\). But maybe I had the sides reversed. Let's assume that the two triangles are similar, with the top triangle having base \( c \) and the bottom triangle (the larger one) having base \( b \), and the legs are \( x \) and \( x + a \). So similarity ratio: \(\frac{x}{x + a}=\frac{c}{b}\), solving for \(x\):

\(bx=cx+ca\)

\(bx - cx=ca\)

\(x(b - c)=ca\)

\(x=\frac{ac}{b - c}\)

Answer:

\(\boldsymbol{\frac{ac}{b - c}}\) (Note: If the proportion was set up incorrectly, but based on the similar triangles with the given labels, this is the solution. If the correct proportion is \(\frac{x}{a}=\frac{c}{b}\), then \(x = \frac{ac}{b}\), but the diagram suggests nested triangles, so the first solution is more likely.)