Chapter 4.5, Problem 10WE

(a)

To determine

Draw an isosceles $\Delta RST$ with $\overline{RT}\cong \overline{ST}$ .Draw

  $\overline{RM}\text{ and }\overline{SN}$ and label their common point O.Now draw $\overline{NM}$ .

Explanation of Solution

Given:

M is the midpoint of $\overline{ST}$ and N is the midpoint of $\overline{RT}$ .

Calculation:

1. Draw an isosceles $\Delta RST$ with $\overline{RT}\cong \overline{ST}$ . M is the midpoint of $\overline{ST}$ and N is the midpoint of $\overline{RT}$ .

  

2. Draw $\overline{RM}\text{ and }\overline{SN}$ and label their common point O.

  

Draw $\overline{NM}$ .

  

(b)

To determine

Name four pairs of congruent triangles.

Answer to Problem 10WE

The four pairs of congruent triangles are :

  $\Delta RTM\text{ and }\Delta STN$

  $\Delta RNM\text{ and }\Delta SMN$

  $\Delta SRN\text{ and }\Delta RSM$

  $\Delta NOR\text{ and }\Delta MOS$

Explanation of Solution

Given:

  $\overline{RT}\cong \overline{ST}$

M is the midpoint of $\overline{ST}$ and N is the midpoint of $\overline{RT}$ .

Calculation:

From part (a) , the triangle is :

  

Since triangle RTS is isosceles and $\overline{RT}\cong \overline{ST}$ ,so, $\angle R\cong \angle S$ .

Also , M and N are the mid points of the sides ST and RT respectively , so , SM = MT and RN = NT .

ST = SM+MT = 2SM = 2MT

RT = RN+NT = 2 RN = 2NT

ST = RT = 2SM = 2RN = 2MT = 2NT

So, SM=RN=MT=NT

  $\overline{SM}\cong \overline{RN}\cong \overline{MT}\cong \overline{NT}$

Also, in the isosceles trapezium MNRS , the diagonals RM and SN are congruent.

  $\overline{RM}\cong \overline{SN}$

Consider $\Delta RTM\text{ and }\Delta STN$ :

  $\overline{RT}\cong \overline{ST}$

  $\angle T\cong \angle T$

  $\overline{TM}\cong \overline{TN}$

So, by SAS Postulate , $\Delta RTM\cong \Delta STN$

Consider $\Delta RNM\text{ and }\Delta SMN$

  $\overline{RN}\cong \overline{SM}$

  $\overline{NM}\cong \overline{MN}$

  $\overline{RM}\cong \overline{SN}$

So, by SSS Postulate , $\Delta RNM\cong \Delta SMN$

Consider $\Delta SRN\text{ and }\Delta RSM$

  $\overline{SR}\cong \overline{RS}$

  $\angle R\cong \angle S$

  $\overline{RN}\cong \overline{SM}$

So, by SAS Postulate , $\Delta SRN\cong \Delta RSM$

Consider $\Delta NOR\text{ and }\Delta MOS$

  $\angle MRN\cong \angle NSM\mathrm{...............}[\Delta RNM\cong \Delta SMN]$

  $\overline{NR}\cong \overline{MS}$

  $\angle RNS\cong \angle SMR\mathrm{...........}[\Delta SRN\cong \Delta RSM]$

So, by ASA Postulate , $\Delta NOR\cong \Delta MOS$

So, the four pairs of congruent triangles are :

  $\Delta RTM\text{ and }\Delta STN$

  $\Delta RNM\text{ and }\Delta SMN$

  $\Delta SRN\text{ and }\Delta RSM$

  $\Delta NOR\text{ and }\Delta MOS$