Question

find the value of x.
(a) triangle with segments 5, 4, 10, x x =
(b) right triangle with segments 6, 6, 10, x x =
(c) right triangle with segments 9, 8, 4, x x =

Explanation:

Part (a)

Step1: Identify similar triangles

The triangles are similar by the Basic Proportionality Theorem (Thales' theorem), so the ratios of corresponding sides are equal. Let the smaller triangle have sides 4, 5, and the larger triangle have sides 10, \(x + 5\)? Wait, no, looking at the sides: the smaller triangle has sides 4 and 5, the larger triangle (including the smaller one) has sides 10 and \(x + 5\)? Wait, no, actually, the segment of length 4 corresponds to 10, and 5 corresponds to \(x + 5\)? Wait, no, maybe the sides are 4 (small) and 10 (large) on one side, 5 (small) and \(x + 5\)? Wait, no, the diagram shows a triangle with a line parallel to the base, so the two triangles are similar. So the ratio of the sides should be equal. So \(\frac{4}{10}=\frac{5}{x + 5}\)? Wait, no, maybe the sides are 4 (small) and 10 (large) on one side, and 5 (small) and \(x\) (large)? Wait, no, let's re-examine. The smaller triangle has sides 4 and 5, the larger triangle (the whole thing) has sides 10 and \(x + 5\)? Wait, no, the diagram: the left side of the smaller triangle is 5, and the left side of the larger triangle (from the vertex to the base) is \(x + 5\)? Wait, no, the left side of the smaller triangle is 5, and the left side of the larger triangle (the segment from the vertex to the base) is \(x\)? Wait, maybe the sides are: the smaller triangle has sides 4 (right) and 5 (left), and the larger triangle (the whole) has right side 10 and left side \(x + 5\)? No, that doesn't make sense. Wait, maybe the two triangles are similar, so the ratio of corresponding sides is equal. So \(\frac{4}{10}=\frac{5}{x}\)? No, that would be if 4 corresponds to 10, and 5 corresponds to \(x\). Wait, let's check the ratios. If the smaller triangle has sides 4 and 5, and the larger triangle (the one with the base) has sides 10 and \(x\), but actually, the line is parallel, so the triangles are similar. So the ratio of the sides of the smaller triangle to the larger triangle is \(\frac{4}{10}=\frac{5}{x}\)? Wait, no, that would be \(\frac{4}{10}=\frac{5}{x}\), then \(4x = 50\), \(x = 12.5\)? Wait, no, maybe I got the sides wrong. Wait, the smaller triangle: top side? No, the diagram shows a triangle with a line parallel to the base, so the two triangles (smaller and larger) are similar. So the ratio of the sides: the side of length 4 in the smaller triangle corresponds to the side of length 10 in the larger triangle, and the side of length 5 in the smaller triangle corresponds to the side of length \(x\) in the larger triangle? Wait, no, maybe the left side of the smaller triangle is 5, and the left side of the larger triangle (from the vertex to the base) is \(x\), and the right side of the smaller triangle is 4, and the right side of the larger triangle is 10. So the ratio is \(\frac{4}{10}=\frac{5}{x}\)? No, that would be \(\frac{4}{10}=\frac{5}{x}\), so \(4x = 50\), \(x = 12.5\). Wait, but let's do it correctly. Let’s denote the smaller triangle as having sides \(a = 4\), \(b = 5\), and the larger triangle as having sides \(A = 10\), \(B = x + 5\)? No, maybe the left side of the smaller triangle is 5, and the left side of the larger triangle (the entire left side) is \(x + 5\), and the right side of the smaller triangle is 4, and the right side of the larger triangle is 10. Wait, no, the right side of the smaller triangle is 4, and the right side of the larger triangle is 10, so the ratio of similarity is \(\frac{4}{10}=\frac{2}{5}\). Then the left side of the smaller triangle is 5, so the left side of the larger triangle (the entire left side) is \(5 \div…

Step1: Identify similar triangles

The two triangles (the smaller one on top and the larger one) are similar because they are both right triangles and share the same angle at the top. The height of the smaller triangle is 6, and the height of the larger triangle is \(6 + 6 = 12\). The base of the larger triangle is 10, and the base of the smaller triangle is \(x\).

Step2: Set up the proportion

Since they are similar, the ratio of height to base is equal. So \(\frac{6}{12}=\frac{x}{10}\).

Step3: Solve for \(x\)

Simplify \(\frac{6}{12}=\frac{1}{2}\), so \(\frac{1}{2}=\frac{x}{10}\)
Cross-multiplying: \(2x = 10\)
\(x = 5\)

Step1: Identify similar triangles

The two right triangles (the larger one and the smaller one) are similar because they share the same angle at the right end. The base of the larger triangle is \(8 + 4 = 12\), and the height is 9. The base of the smaller triangle is 4, and the height is \(x\).

Step2: Set up the proportion

Since they are similar, the ratio of base to height is equal. So \(\frac{12}{9}=\frac{4}{x}\).

Step3: Solve for \(x\)

Cross-multiplying: \(12x = 9 \times 4\)
\(12x = 36\)
\(x = \frac{36}{12} = 3\)

Answer:

(a):
\(x = 12.5\)

Part (b)