Answer:
[tex]x = 23[/tex]
[tex]ECH = 28[/tex]
[tex]HCD = 62[/tex]
[tex]GCF = 28[/tex]
[tex]ECG = 152[/tex]
[tex]GCD = 118[/tex]
Step-by-step explanation:
Given
[tex]ECH = x + 5[/tex]
[tex]HCD = 3x - 7[/tex]
Solving (a): The value of x
Since CD is perpendicular to EF, then the following relationship exists
[tex]ECH + HCD = ECD[/tex]
Where ECD = 90° and DCF = 90°
Substitute values for ECH, HCD and ECD
[tex]x + 5 + 3x - 7 = 90[/tex]
Collect like terms
[tex]x + 3x = 90 + 7 - 5[/tex]
[tex]4x = 92[/tex]
Divide both sides by 4
[tex]x = 23[/tex]
Solving (b): The value of ECH
Given that
[tex]ECH = x + 5[/tex]
Substitute 23 for x
[tex]ECH = 23 + 5[/tex]
[tex]ECH = 28[/tex]
Solving (c): The value of HCD
Given that
[tex]HCD = 3x - 7[/tex]
Substitute 23 for x
[tex]HCD = 3 * 23 - 7[/tex]
[tex]HCD = 69 - 7[/tex]
[tex]HCD = 62[/tex]
Solving (d): Solving GCF
GCF and ECH are opposites
Hence,
[tex]GCF = ECH = 28[/tex]
Solving (e): The value of ECG
[tex]ECG + ECH = 180[/tex]
Make ECG the subject of formula
[tex]ECG = 180 - ECH[/tex]
Substitute 28 for ECH
[tex]ECG = 180 - 28[/tex]
[tex]ECG = 152[/tex]
Solving (f): The value of GCD
[tex]GCD = GCF + DCF[/tex]
Recall that DCF = 90 and GCF = 28
[tex]GCD = 90 + 28[/tex]
[tex]GCD = 118[/tex]
