Question

try again
your answer is incorrect.
find the length x.
(image of a right triangle with a smaller right triangle inside, showing segments 7.5, 4.5, 7.5, and x, and a previous answer of 4.5 marked incorrect)

Explanation:

Step1: Identify Similar Triangles

The two triangles (smaller and larger) are similar by AA (Angle - Angle) similarity (both have a right angle and share the same non - right angle). For similar triangles, the ratios of corresponding sides are equal. Let the length of the hypotenuse of the smaller triangle be \(7.5\) and the hypotenuse of the larger triangle be \(7.5 + 7.5=15\). The corresponding side of the smaller triangle to \(x\) is \(4.5\).

Step2: Set Up Proportion

We set up the proportion \(\frac{4.5}{x}=\frac{7.5}{15}\). Cross - multiply: \(7.5x = 4.5\times15\).

Step3: Solve for \(x\)

First, calculate \(4.5\times15 = 67.5\). Then, \(x=\frac{67.5}{7.5}\). Dividing \(67.5\) by \(7.5\) gives \(x = 9\). Wait, maybe the triangles are similar in another way. Let's re - examine the diagram. The two segments on the hypotenuse are both \(7.5\), so the ratio of the sides of the smaller triangle to the larger triangle is \(\frac{7.5}{7.5 + 7.5}=\frac{1}{2}\)? No, wait, the horizontal side of the smaller right - angled trapezoid? No, the two right - angled triangles: the smaller one (top) and the larger one (bottom - big). Wait, the smaller triangle has a horizontal side of \(4.5\) and the larger one has a horizontal side of \(x\). The ratio of their hypotenuses: the smaller hypotenuse is \(7.5\), the larger hypotenuse is \(7.5+7.5 = 15\). So the ratio of similarity is \(\frac{7.5}{15}=\frac{1}{2}\). Wait, but if the smaller triangle's horizontal side is \(4.5\), then the larger one's horizontal side \(x\) should be \(4.5\times2=9\)? But maybe the diagram is of two similar triangles with the smaller triangle having a side of \(4.5\) and the ratio of sides is \(\frac{7.5}{7.5 + 7.5}=\frac{1}{2}\)? No, perhaps the correct proportion is \(\frac{4.5}{x}=\frac{7.5}{7.5 + 7.5}\)? No, that would be wrong. Wait, maybe the two triangles (the upper small triangle and the lower large triangle) are similar, with the upper triangle having a side of \(4.5\) and the lower triangle having a side of \(x\), and the ratio of their hypotenuses is \(\frac{7.5}{7.5+7.5}=\frac{1}{2}\)? No, that would make \(x = 9\). But maybe I misread the diagram. Wait, the diagram shows a right - angled triangle with a line parallel to the base (the side \(x\)) cutting the triangle into a smaller right - angled triangle and a trapezoid. The two segments on the hypotenuse are both \(7.5\), so the ratio of the sides of the smaller triangle to the larger triangle is \(\frac{7.5}{7.5 + 7.5}=\frac{1}{2}\). So if the smaller triangle's base (parallel to \(x\)) is \(4.5\), then the larger triangle's base \(x\) is \(4.5\times2 = 9\). Wait, but maybe the initial answer of \(4.5\) was wrong, and the correct answer is \(9\). Wait, let's do it again. Let the smaller triangle (top) have sides: let's say the vertical side is \(a\), horizontal side \(4.5\), hypotenuse \(7.5\). The larger triangle (bottom - big) has vertical side \(a + b\), horizontal side \(x\), hypotenuse \(7.5+7.5 = 15\). Since they are similar, \(\frac{4.5}{x}=\frac{7.5}{15}\), so \(x=\frac{4.5\times15}{7.5}=\frac{67.5}{7.5}=9\).

Answer:

\(9\)