Question

3 numeric 2 points 6. 8 16 -2 + 6x 42 find x. give only the number.

Explanation:

Step1: Use the mid - segment theorem

The mid - segment of a triangle is parallel to the third side and half its length. Here, the length of the mid - segment is related to the base. The ratio of the sides of the smaller triangle to the larger triangle is \(\frac{8}{8 + 16}=\frac{8}{24}=\frac{1}{3}\). But using the mid - segment formula (if we consider the correct proportion), we know that if a line segment divides two sides of a triangle proportionally, then it is parallel to the third side. Here, since the segments on the two sides are in proportion (the marks indicate they are divided proportionally), we can set up the equation \(\frac{- 2+6x}{42}=\frac{16 + 8}{16+8 + 8}\) (using the side - splitter theorem). Wait, more simply, using the mid - segment property (assuming the line is a mid - segment, from the red marks which usually denote mid - points in a basic proportion problem). The length of the mid - segment (if we assume the line divides the two sides into equal parts) should satisfy \(-2 + 6x=\frac{42}{2}\) (mid - segment is half of the base when it connects mid - points).

Step2: Solve the equation

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Wait, no, let's re - check. If we use the basic proportionality theorem (Thales' theorem). Let the two sides of the triangle be \(a = 8+16=24\) and \(b\) (the other side). The line divides the sides proportionally. If we assume the line is parallel to the base (from the red marks which are similar to mid - point marks in a basic problem setup in geometry textbooks for teaching the mid - segment or basic proportionality). The length of the line segment (assuming it is a mid - segment, i.e., the ratio of the divided side is \(1:1\)) then \(-2+6x = 21\) (since mid - segment length \(=\frac{1}{2}\times\) base length).

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Wait, no! Wait, the base is \(42\). If the line is a mid - segment (connecting mid - points of two sides), then its length is \(\frac{42}{2}=21\). So we set \(-2 + 6x=21\)

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No, wait, wrong approach. Let's use the side - splitter theorem. The ratio of the segments of one side is \(\frac{8}{16}=\frac{1}{2}\). Then the ratio of the line segment to the base is \(\frac{1}{1 + 2}=\frac{1}{3}\) (if we consider the similar triangles). Wait, no, if two sides are divided proportionally (\(\frac{8}{16}=\frac{1}{2}\)), then the line is parallel to the base. Let the length of the line segment be \(l\). By the property of similar triangles (the small triangle and the large triangle), \(\frac{l}{42}=\frac{8}{8 + 16}=\frac{8}{24}=\frac{1}{3}\). But \(l=-2+6x\). So \(-2+6x=\frac{42}{3}=14\)

Step2: Solve for \(x\)

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Answer:

\(\frac{8}{3}\)