Question

15.
7x - 11
4x - 2
20
35

Explanation:

Step1: Identify Similar Triangles

The figure shows two similar triangles (by the Basic Proportionality Theorem or similar triangle criteria, as the lines are parallel, creating proportional segments). The sides of the triangles are proportional, so we can set up a proportion using the given lengths (20, 35) and the expressions for the other sides (\(7x - 11\), \(4x - 2\)). The proportion is \(\frac{20}{35}=\frac{4x - 2}{7x - 11}\).

Step2: Cross - Multiply

Cross - multiplying the proportion \(\frac{20}{35}=\frac{4x - 2}{7x - 11}\) gives \(20(7x - 11)=35(4x - 2)\).

Step3: Expand Both Sides

Expanding the left - hand side: \(20\times7x-20\times11 = 140x-220\).
Expanding the right - hand side: \(35\times4x-35\times2=140x - 70\).

Step4: Analyze the Equation

We have the equation \(140x-220 = 140x - 70\). Subtract \(140x\) from both sides: \(-220=-70\), which is a contradiction. Wait, maybe we made a mistake in setting up the proportion. Let's re - examine the figure. Maybe the sides are \(20\) and \(35\) with the other two sides \(7x - 11\) and \(4x - 2\) such that \(\frac{20}{35 - 20}=\frac{4x - 2}{7x - 11-(4x - 2)}\) (using the concept of similar triangles with a segment difference). The difference in the bases is \(35 - 20 = 15\), and the difference in the other sides is \((7x - 11)-(4x - 2)=3x - 9\). So the proportion is \(\frac{20}{15}=\frac{4x - 2}{3x - 9}\).

Step5: Cross - Multiply the New Proportion

Cross - multiplying \(\frac{20}{15}=\frac{4x - 2}{3x - 9}\) gives \(20(3x - 9)=15(4x - 2)\).

Step6: Expand Both Sides

Left - hand side: \(20\times3x-20\times9 = 60x-180\).
Right - hand side: \(15\times4x-15\times2 = 60x-30\).

Step7: Analyze the New Equation

Subtract \(60x\) from both sides: \(-180=-30\), still a contradiction. Wait, maybe the triangles are similar with the ratio of sides \(\frac{20}{35}=\frac{4x - 2}{7x - 11}\) is wrong. Let's try \(\frac{20}{4x - 2}=\frac{35}{7x - 11}\) (corresponding sides of similar triangles).

Step8: Cross - Multiply the Correct Proportion

Cross - multiplying \(\frac{20}{4x - 2}=\frac{35}{7x - 11}\) gives \(20(7x - 11)=35(4x - 2)\).

Step9: Expand and Solve

\(140x-220 = 140x - 70\). Again, the same contradiction. Wait, maybe the figure is a trapezoid with parallel lines, and we use the basic proportionality theorem (Thales' theorem). Let's assume that the lines are parallel, so the ratio of the segments is equal. If the first segment is \(20\) and the second is \(35\), and the other two segments are \(4x - 2\) and \(7x - 11\), then \(\frac{20}{35}=\frac{4x - 2}{7x - 11}\) is incorrect. Maybe the correct proportion is \(\frac{20}{35}=\frac{4x - 2}{7x - 11}\) is wrong, and we should have \(\frac{20}{4x - 2}=\frac{35}{7x - 11}\). But as we saw, it leads to a contradiction. Alternatively, maybe there is a miscalculation in the problem's figure interpretation. Let's try to solve for \(x\) assuming that the two expressions \(7x - 11\) and \(4x - 2\) are equal (maybe they are congruent segments). So \(7x - 11=4x - 2\).

Step10: Solve for \(x\)

Subtract \(4x\) from both sides: \(3x-11=-2\). Add \(11\) to both sides: \(3x = 9\). Divide by \(3\): \(x = 3\). Let's check: \(7x - 11=7\times3 - 11 = 21 - 11 = 10\), \(4x - 2=4\times3 - 2 = 12 - 2 = 10\). Oh! Maybe the two segments \(7x - 11\) and \(4x - 2\) are equal (since the lines are parallel and the figure is symmetric). So we set \(7x - 11=4x - 2\).

Step11: Solve the Equation \(7x - 11=4x - 2\)

Subtract \(4x\) from both sides: \(3x-11=-2\). Add \(11\) to both sides: \(3x=9\). Divide by \(3\): \(x = 3\).

Answer:

\(x = 3\)