Chapter 15.3, Problem 15PSB

(a)

To determine

To Find: The longer side in the given triangle.

Answer to Problem 15PSB

  $\overline{AC}$

Explanation of Solution

Given:

  $\begin{array}{l}\to \overline{AD}\text{ is median of }\Delta \text{ABC}\\ \to m\angle ADC=2x+35\\ \to m\angle ADB=5x-65\end{array}$

Formula Used:

The Hinge Theorem: It states that if two sides of two triangles are congruent and angle of one is greater than the angle of second then the third side of the triangle will be greater than the another.

Calculation:

  

  $\begin{array}{l}\to \text{In }\Delta ADC\text{ and }\Delta ADB\\ \text{⇒}AD=AD\left(\text{Reflexive property of congruence}\right)\\ \text{⇒}m\angle ADC+m\angle ADB=180\left(\text{Linear pair}\right)\\ \text{⇒}\left(2x+35\right)+\left(5x-65\right)=180\left(\text{Given}\right)\\ \text{⇒}7x-30=180\\ \text{⇒}7x-30+30=180+30\left(\text{add both side by }30\right)\\ \text{⇒}\frac{7x}{7}=\frac{210}{7}\left(\text{divide both side by 7}\right)\\ \text{⇒}x=30\\ \text{⇒}m\angle ADC=2x+35=2\left(30\right)+35=95\\ \text{⇒}m\angle ADB=5x-65=5\left(30\right)-65=85\\ \text{⇒}\angle ADC>\angle ADB\\ \text{⇒}\overline{AC}>\overline{AB}\left(\text{SAS inequality theorem}\right)\end{array}$

Conclusion:

Thus the longer side is $\overline{AC}$

(b)

To determine

To Find: The larger angle in the given triangle.

Answer to Problem 15PSB

  $\angle C$

Explanation of Solution

Given:

  $\begin{array}{l}\to \overline{AD}\text{ is median of }\Delta \text{ABC}\\ \to m\angle ADC=2x+35\\ \to m\angle ADB=5x-65\end{array}$

Formula Used:

The Hinge Theorem: It states that if two sides of two triangles are congruent and angle of one is greater than the angle of second then the third side of the triangle will be greater than the another.

Calculation:

  

  $\begin{array}{l}\to \text{In }\Delta ADC\text{ and }\Delta ADB\\ \text{⇒}AD=AD\left(\text{Reflexive property of congruence}\right)\\ \text{⇒}m\angle ADC+m\angle ADB=180\left(\text{Linear pair}\right)\\ \text{⇒}\left(2x+35\right)+\left(5x-65\right)=180\left(\text{Given}\right)\\ \text{⇒}7x-30=180\\ \text{⇒}7x-30+30=180+30\left(\text{add both side by }30\right)\\ \text{⇒}\frac{7x}{7}=\frac{210}{7}\left(\text{divide both side by 7}\right)\\ \text{⇒}x=30\\ \text{⇒}m\angle ADC=2x+35=2\left(30\right)+35=95\\ \text{⇒}m\angle ADB=5x-65=5\left(30\right)-65=85\\ \text{⇒}\angle ADC>\angle ADB\\ \text{⇒}\overline{AC}>\overline{AB}\left(\text{SAS inequality theorem}\right)\\ \text{⇒}\angle B<\angle C\left(\text{SAS inequality theorem}\right)\end{array}$

Conclusion:

Thus the larger angle is $\angle C$