Question

1. find the value of x. 7. solve for y. 8. find the value of x. 9. solve for the variable.

Explanation:

Step1: Apply the triangle - proportionality theorem

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. For the first triangle with sides 5, 15, \(x + 3\) and 25, we have the proportion \(\frac{5}{15}=\frac{x + 3}{25}\).

Step2: Cross - multiply

Cross - multiplying the proportion \(\frac{5}{15}=\frac{x + 3}{25}\) gives us \(5\times25=15\times(x + 3)\). So, \(125 = 15x+45\).

Step3: Solve for \(x\)

Subtract 45 from both sides: \(125−45 = 15x\), which simplifies to \(80 = 15x\). Then \(x=\frac{80}{15}=\frac{16}{3}\).

Step4: For the second triangle

Using the same triangle - proportionality theorem, with sides 7, 10, 25 and \(y\), we have the proportion \(\frac{7}{10}=\frac{25}{y}\).

Step5: Cross - multiply for \(y\)

Cross - multiplying gives \(7y=10\times25\), so \(7y = 250\).

Step6: Solve for \(y\)

\(y=\frac{250}{7}\)

Answer:

\(x=\frac{16}{3}\), \(y=\frac{250}{7}\)