Chapter 7.2, Problem 13PSB

a

To determine

To prove : $\Delta HRJ$ is isosceles.

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{OH}\cong \overrightarrow{MJ}$

  $\angle OHJ=\angle MJH$

  $\angle HPJ=\angle JKH$

Formula used : If two triangles have two congruent angles and the side between them is also congruent, then the triangles are also said to be congruent

Proof : We know that in triangles HPK and JKH

  $\angle OHJ=\angle MJH$

  $\angle HPJ=\angle JKH$

Thus, as the sum of the angles of the triangles is same, we can say that

  $\angle RJH=\angle RHJ$

Now, if $\angle RJH=\angle RHJ$ , then we can say that

  $\Delta HRJ$ is isosceles

b

To determine

  $\overrightarrow{HP}=\overrightarrow{JK}$ is to be proved

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{OH}\cong \overrightarrow{MJ}$

  $\angle OHJ=\angle MJH$

  $\angle HPJ=\angle JKH$

Formula used : If two triangles have two congruent angles and the side between them is also congruent, then the triangles are also said to be congruent

Proof : We know that in triangles HPK and JKH

  $\angle OHJ=\angle MJH$

  $\angle HPJ=\angle JKH$

Thus, as the sum of the angles of the triangles is same, we can say that

  $\angle RJH=\angle RHJ$

Thus, if all three angles of HPK and JKH are congruent, and also

HJ is the common chord

Hence, we can say that

  $\Delta HPJ\cong \Delta JKH$

Now, by virtue of congruency, we can say that

  $\overrightarrow{HP}=\overrightarrow{JK}$

c

To determine

To prove : R is equidistant from O and M.

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{OH}\cong \overrightarrow{MJ}$

  $\angle OHJ=\angle MJH$

  $\angle HPJ=\angle JKH$

Formula used : If two triangles have two congruent angles and the side between them is also congruent, then the triangles are also said to be congruent

Proof : We know that in triangles ORP and MRK

  $\angle OPR=\angle MKR$

  $\overrightarrow{PO}\cong \overrightarrow{KM}$

  $\overrightarrow{PR}\cong \overrightarrow{RK}$

Thus, by virtue of congruency of the triangles, we can say that

  $\Delta ROP=\Delta RMK$

Now, if $\Delta ROP=\Delta RMK$ , then we can say that

  $\overrightarrow{RO}\cong \overrightarrow{RM}$

Hence, given distance is proved