Question

1. 48 48 24 -6 + 4x 40 40 35 10) 2 4 4 7 a) 8 b) 36 c) 14 d) 16

Explanation:

Step1: Use the property of similar polygons

For similar polygons, the ratios of corresponding sides are equal.
$$\frac{48}{40}=\frac{- 6 + 4x}{35}$$

Step2: Cross - multiply

Cross - multiplying gives \(48\times35 = 40\times(-6 + 4x)\).
First, calculate \(48\times35=1680\) and \(40\times(-6 + 4x)=-240+160x\). So, \(1680=-240 + 160x\).

Step3: Solve for \(x\)

Add \(240\) to both sides: \(1680 + 240=160x\), i.e., \(1920 = 160x\). Then, divide both sides by \(160\): \(x=\frac{1920}{160}=12\).

For problem 10:

Step1: Use the basic proportionality theorem (Thales' theorem)

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Let the unknown side be \(y\).
We have \(\frac{2}{2 + 4}=\frac{4}{4 + y}\) (using the ratio of segments of two sides of the triangle).

Step2: Simplify the left - hand side

\(\frac{2}{6}=\frac{1}{3}\). So, \(\frac{1}{3}=\frac{4}{4 + y}\).

Step3: Cross - multiply

Cross - multiplying gives \(4 + y=12\). Then, subtract \(4\) from both sides: \(y = 8\).

Answer:

For problem 8, \(x = 12\). For problem 10, the answer is A. 8.