Question

multi-step find the side lengths of each triangle.
18.

19.

Explanation:

Problem 18:

Step1: Identify the triangle type

The triangle has two equal - marked sides, so it is an isosceles triangle. So we can set \(z + 5=3z - 1\) (since the two marked sides are equal).

Step2: Solve for \(z\)

\[

$$\begin{align*} z+5&=3z - 1\\ 5 + 1&=3z - z\\ 6&=2z\\ z&=3 \end{align*}$$

\]

Step3: Find the side lengths

• For the side \(z + 5\): Substitute \(z = 3\), we get \(3+5 = 8\).
• For the side \(3z - 1\): Substitute \(z = 3\), we get \(3\times3-1=9 - 1 = 8\).
• For the side \(4z - 4\): Substitute \(z = 3\), we get \(4\times3-4=12 - 4 = 8\). So the triangle is equilateral with side length 8.

Problem 19:

Step1: Identify the triangle type

The triangle is a right - isosceles triangle (the two legs are marked as equal). So we set \(2x + 6.8=8x + 1.4\).

Step2: Solve for \(x\)

\[

$$\begin{align*} 2x+6.8&=8x + 1.4\\ 6.8-1.4&=8x - 2x\\ 5.4&=6x\\ x& = 0.9 \end{align*}$$

\]

Step3: Find the side lengths

• For the leg \(2x + 6.8\): Substitute \(x = 0.9\), we get \(2\times0.9+6.8 = 1.8+6.8=8.6\).
• For the leg \(8x + 1.4\): Substitute \(x = 0.9\), we get \(8\times0.9+1.4=7.2 + 1.4 = 8.6\).

Answer:

(Problem 18):
The side lengths of the triangle are 8, 8, 8.