Answer:
Part 1) The length of DC is [tex]13\ units[/tex]
Part 2) The measures of the angles in the isosceles triangle are
The base angles are 67.4° and the vertex angle is 45.2°
Step-by-step explanation:
step 1
In the right triangle BDC Find the length of DC
Applying the Pythagoras Theorem
[tex]DC^{2}=BC^{2} +BD^{2}[/tex]
we have
[tex]BC=AB=10/2=5\ units[/tex]
[tex]BD=12\ units[/tex]
substitute
[tex]DC^{2}=5^{2} +12^{2}[/tex]
[tex]DC^{2}=169[/tex]
[tex]DC=13\ units[/tex]
step 2
Find the measures of internal angles in the isosceles triangle ABC
we know that
∠DAC=∠DCA ------> base angles
∠ADC ------> vertex angle
Find the measure of angle DCA
In the right triangle BDC
sin(∠DCA)=BD/DC
substitute the values
sin(∠DCA)=12/13
∠DCA=arcsin(12/13)=67.4°
so
∠DAC=∠DCA=67.4°
Find the measure of angle ∠ADC
Remember that the sum of the internal angles of a triangle must be equal to 180 degrees
so
∠DAC+∠DCA+∠ADC=180°
substitute
67.4°+67.4°+∠ADC=180°
∠ADC=180°-134.8°=45.2°
