ABCD is a square and AQ is perpendicular to PR. PQR and ADR are straight lines. DQ : DC = 1 : 2. Show that triangle DQR is congruent to triangle CQP.
Give a reason for each statement you make. Answer

Determining triangles Ratios rational Properties shapes Angle relationships 中文

It won't just hand you the answer — it asks where you're stuck.How the checker works

Worked solution

6

1. Show that the corresponding side lengths $DQ$ and $QC$ are equal.

Given $DQ:DC=1:2$ ,
$DQ=\frac12DC.$
Since $Q$ lies on $DC$ ,
$DQ+QC=DC.$
Hence
$QC=DC-DQ=DC-\frac12DC=\frac12DC.$
Therefore,
$DQ=QC.$

2. Show that two pairs of corresponding angles are equal and conclude congruence.

In the square, $AD\perp DC$ and $BC\perp DC$ . Since $D,A,R$ are collinear and $C,B,P$ are collinear,
$\angle QDR=\angle QCP=90^\circ.$
Also, since $D,Q,C$ are collinear and $R,Q,P$ are collinear,
$\angle DQR=\angle CQP$
vertically opposite angles.

Thus, with $DQ=QC$ ,
$\triangle DQR\cong\triangle CQP$
by ASA.

Answer

Since $DQ:DC=1:2$ , $DQ=\dfrac12DC$ . As $Q$ lies on $DC$ , $DQ+QC=DC$ , so $QC=\dfrac12DC$ . Hence $DQ=QC$ .

Also, $\angle QDR=\angle QCP=90^\circ$ since $AD\perp DC$ and $BC\perp DC$ in the square.

$\angle DQR=\angle CQP$ since they are vertically opposite angles, as $D,Q,C$ are collinear and $R,Q,P$ are collinear.

Therefore, $\triangle DQR\cong\triangle CQP$ by ASA.

AI analysis — may contain mistakes.

Worked this one out on paper? Snap your steps and Gigle's AI math checker will circle the step where it went wrong.

ABCD is a square and AQ is perpendicular to PR. PQR and ADR are straight lines. DQ : DC = 1 : 2. Show that triangle DQR is congruent to triangle CQP. Give a reason for each statement you make. Answer

Worked solution

6

More questions on this topic

ABCD is a square and AQ is perpendicular to PR. PQR and ADR are straight lines. DQ : DC = 1 : 2. Show that triangle DQR is congruent to triangle CQP.
Give a reason for each statement you make. Answer