TEOREMA DE TALES $\newline$ Se duas retas são transversais de um conjunto de retas paralelas, então a razão entre dois segmentos quaisquer de uma delas é igual à razäo entre os segmentos correspondentes da outra.

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Estimate products of mixed numbers

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Q. TEOREMA DE TALES

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Se duas retas são transversais de um conjunto de retas paralelas, então a razão entre dois segmentos quaisquer de uma delas é igual à razäo entre os segmentos correspondentes da outra.

Thales' Theorem Explanation: Thales' Theorem states that if two lines are intersecting a set of parallel lines, then the ratio of any two segments on one of the intersecting lines is equal to the ratio of the corresponding segments on the other intersecting line. To illustrate this, let's consider two parallel lines, line A and line B, and two transversal lines, line $1$ and line $2$ , intersecting them. If we take segments on line $1$ , say $a$ and $b$ , and the corresponding segments on line $2$ , say $c$ and $d$ , then according to Thales' Theorem, the ratio $\frac{a}{b}$ should be equal to the ratio $\frac{c}{d}$ .
Identifying Corresponding Segments: To apply Thales' Theorem, we need to identify the corresponding segments on the transversal lines. Let's say we have segments $a_1$ and $b_1$ on line $1$ , and their corresponding segments on line $2$ are $a_2$ and $b_2$ . We can then write the equation $\frac{a_1}{b_1} = \frac{a_2}{b_2}$ to represent the relationship between these segments.
Plugging in Measurements: If we have specific measurements for these segments, we can plug them into the equation to find the unknown segment or to verify the relationship. For example, if $a_1 = 3\, \text{cm}$ , $b_1 = 6\, \text{cm}$ , $a_2 = 4\, \text{cm}$ , and we want to find $b_2$ , we would set up the equation $\frac{3}{6} = \frac{4}{b_2}$ and solve for $b_2$ .
Cross-Multiplying: Solving the equation $\frac{3}{6} = \frac{4}{b^2}$ , we cross-multiply to get $3 \cdot b^2 = 4 \cdot 6$ . This simplifies to $3 \cdot b^2 = 24$ .
Solving for Unknown Segment: Dividing both sides of the equation by $3$ to solve for $b_2$ , we get $b_2 = \frac{24}{3}$ , which simplifies to $b_2 = 8\,\text{cm}$ .