Question

find the lengths of all three sides of each triangle
see example 3
20.
21.
use the figure shown for exercises 22 and 23.

Explanation:

Step1: Identify equal - length sides in triangle 20

In triangle \(DEF\), since the angles opposite to sides \(EF\) and \(DF\) are equal (marked with the same arc), then \(EF = DF\).
So, \(4x−12=3x + 5\).

Step2: Solve the equation for \(x\) in triangle 20

Subtract \(3x\) from both sides: \(4x-3x−12=3x - 3x+5\), which gives \(x−12 = 5\).
Add 12 to both sides: \(x=5 + 12=17\).

Step3: Find the lengths of the sides in triangle 20

For side \(EF\): \(4x−12=4\times17−12=68 - 12 = 56\).
For side \(DF\): \(3x + 5=3\times17+5=51 + 5=56\).
The third side \(ED = 20\).

Step4: Identify equal - length sides in triangle 21

In triangle \(GHJ\), since the angles opposite to sides \(GH\) and \(GJ\) are equal (marked with the same arc), then \(GH=GJ\).
So, \(4x + 1=45\).

Step5: Solve the equation for \(x\) in triangle 21

Subtract 1 from both sides: \(4x+1 - 1=45 - 1\), which gives \(4x=44\).
Divide both sides by 4: \(x = 11\).

Step6: Find the lengths of the sides in triangle 21

For side \(GJ\): \(4x + 1=4\times11+1=44 + 1=45\).
For side \(HJ\): \(6x+3=6\times11+3=66 + 3=69\).

Answer:

• Triangle 20: The side lengths are \(20\), \(56\), \(56\).
• Triangle 21: The side lengths are \(45\), \(45\), \(69\).