Question

each pair of triangles is similar. find the value of x.
5.
6.
7.
8.

Explanation:

Problem 5:

Step1: Identify similarity ratio

The sides of similar triangles are proportional. The corresponding sides are 6 (large) and 4 (small), and 2 (small) and \( x \) (large). The ratio of large to small is \( \frac{6}{4}=\frac{3}{2} \).

Step2: Set up proportion

Using the ratio, \( \frac{x}{2}=\frac{6}{4} \). Cross - multiply: \( 4x = 6\times2 \).

Step3: Solve for \( x \)

\( 4x=12 \), so \( x = \frac{12}{4}=3 \).

Problem 6:

Step1: Determine corresponding sides

The sides of the large triangle: \( 4 + 6=10 \) (left side) and \( x\) (right side). The small triangle has sides 6 (top) and 9 (top - right). Wait, actually, the line divides the triangle proportionally. So \( \frac{4}{4 + 6}=\frac{x - 9}{x}\)? No, better: the two triangles (small and large) are similar. The sides of the small triangle: 6 and 9. The sides of the large triangle: \( 4+6 = 10\) and \( x\). Wait, no, the segments: the left side of the large triangle is \( 4 + 6=10 \), and the left side of the small triangle is 6. The right side of the small triangle is 9, and the right side of the large triangle is \( x\). So the proportion is \( \frac{6}{10}=\frac{9}{x}\)? No, wait, the line is parallel, so by the Basic Proportionality Theorem (Thales' theorem), \( \frac{4}{4 + 6}=\frac{x - 9}{x}\)? No, let's re - examine. The small triangle has sides 6 (top) and 9 (top - right), and the large triangle (including the small one) has left side \( 4+6 = 10\) and right side \( x\). Wait, actually, the two triangles (the small one and the large one) are similar. So the ratio of corresponding sides: \( \frac{6}{6 + 4}=\frac{9}{x}\)? No, \( \frac{6}{4 + 6}=\frac{9}{x}\) → \( \frac{6}{10}=\frac{9}{x}\) → \( 6x=90\) → \( x = 15\). Wait, let's do it correctly. The sides of the small triangle: 6 (left - top) and 9 (right - top). The sides of the large triangle: \( 4 + 6=10\) (left) and \( x\) (right). Since they are similar, \( \frac{6}{10}=\frac{9}{x}\) → cross - multiply: \( 6x=90\) → \( x = 15\).

Problem 7:

Step1: Recall geometric mean in right triangles

In a right triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse. So \( x^{2}=72\times128\).

Step2: Calculate \( 72\times128\)

\( 72\times128=(70 + 2)\times128=70\times128+2\times128=8960+256 = 9216\).

Step3: Find \( x \)

\( x=\sqrt{9216}=96\) (since length is positive).

Problem 8:

Answer:

s:
Problem 5: \( \boldsymbol{3}\)
Problem 6: \( \boldsymbol{15}\)
Problem 7: \( \boldsymbol{96}\)
Problem 8: \( \boldsymbol{\frac{16}{3}}\) (or \( 5\frac{1}{3}\))