Question

1.
32
x
24
9

Explanation:

Step1: Identify Similar Triangles

The line segment in the triangle is parallel to the base, so the two triangles are similar by the Basic Proportionality Theorem (Thales' theorem). For similar triangles, the ratios of corresponding sides are equal. Let the length of the entire side (including \(x\)) be \(32 + x\)? Wait, no, wait. Wait, the side with length 32: the smaller triangle has a side of length 32, and the larger triangle has a side of length \(32 + x\)? Wait, no, looking at the diagram: the base of the smaller triangle is 24, the base of the larger triangle is \(24 + 9 = 33\)? Wait, no, maybe the sides are proportional. Wait, the two triangles: the smaller triangle has sides 32 (left side), 24 (base), and the larger triangle has left side \(32 + x\)? No, wait, maybe the left side of the smaller triangle is 32, and the left side of the larger triangle is \(32 + x\)? Wait, no, the diagram shows: the left side of the smaller triangle is 32, and there's a segment \(x\) above it, so the total left side of the larger triangle is \(x + 32\)? Wait, no, maybe the two triangles: the smaller one has base 24, and the larger one has base \(24 + 9 = 33\), and the left side of the smaller is 32, left side of the larger is \(32 + x\)? No, that doesn't make sense. Wait, maybe the correct proportion is \(\frac{32}{32 + x}=\frac{24}{24 + 9}\)? Wait, no, maybe the line is parallel to the right side? No, the arrows are on the two sides, so the line is parallel to the base, so the triangles are similar. So the ratio of the left side of the smaller triangle to the left side of the larger triangle is equal to the ratio of the base of the smaller triangle to the base of the larger triangle. Wait, the base of the smaller triangle is 24, the base of the larger triangle is \(24 + 9 = 33\). The left side of the smaller triangle is 32, the left side of the larger triangle is \(32 + x\)? No, that would mean \(x\) is the extension, but maybe the left side of the larger triangle is \(32\), and the left side of the smaller triangle is \(32 - x\)? No, the diagram shows \(x\) above the 32 segment. Wait, maybe I got the proportion wrong. Let's re-examine: the two triangles are similar, so the ratio of corresponding sides is equal. So the side of length 32 (in the smaller triangle) corresponds to the side of length \(32 + x\) (in the larger triangle), and the base of the smaller (24) corresponds to the base of the larger (24 + 9 = 33). Wait, no, that would be \(\frac{32}{32 + x}=\frac{24}{33}\). Let's solve that: cross-multiplying, \(32 \times 33 = 24 \times (32 + x)\). \(1056 = 768 + 24x\). Then \(24x = 1056 - 768 = 288\), so \(x = 12\). Wait, that makes sense. Let's check: \(\frac{32}{32 + 12}=\frac{32}{44}=\frac{8}{11}\), and \(\frac{24}{24 + 9}=\frac{24}{33}=\frac{8}{11}\). Yes, that works. So the correct proportion is \(\frac{32}{32 + x}=\frac{24}{24 + 9}\).

Step1: Set up the proportion

Since the triangles are similar, the ratio of corresponding sides is equal. So:
\[
\frac{32}{32 + x} = \frac{24}{24 + 9}
\]

Step2: Simplify the right-hand side

Calculate \(24 + 9 = 33\), so:
\[
\frac{32}{32 + x} = \frac{24}{33}
\]

Step3: Cross-multiply

\[
32 \times 33 = 24 \times (32 + x)
\]

Step4: Calculate left-hand side

\(32 \times 33 = 1056\)

Step5: Expand the right-hand side

\(24 \times 32 + 24x = 768 + 24x\)

Step6: Solve for \(x\)

\[
1056 = 768 + 24x
\]
Subtract 768 from both sides:
\[
1056 - 768 = 24x
\]
\[
288 = 24x
\]
Divide both sides by 24:
\[
x = \frac{288}{24} = 12
\]

Answer:

\(x = 12\)