1. ## [SOLVED] quadrilateral QABR

Hi, This question is from the SAT OG Book on page 746 # 12.

Note: Figure not drawn to scale

In the figure above, points P, A and B are equally spaced on line L and points P, Q, and R are equally spaced on line M. If PB=4, PR=6, and AQ=4, what is the perimeter of quadrilateral QABR?

I know from the information that PA=AB=2, and that PQ=QR=3. Therefore I know the sides AB=2, AQ=4, QR=3. How do I find BR?

2. Hello, fabxx!

In the figure below, points $P, A, B$ are equally spaced,
and points $P, Q, R$ are equally spaced.
If $PB=4,\;PR=6,\;AQ=4$, what is the perimeter of quadrilateral $QABR$ ?

Rotate the diagram 90°.
Code:
                      P
o
*  *
3 *     * 2
*        *
Q o * * * * * o A
*       4      *
3  *                 * 2
*                    *
R o * * * * * * * * * * * o B
?

We have: $\Delta PBR$

$QA$ joins the midpoints of two sides.

Hence, $QA \parallel RB$ and $QA = \frac{1}{2}(RB)$

Therefore: . $RB \:=\:8$

3. Originally Posted by Soroban
Hello, fabxx!

Rotate the diagram 90°.
Code:
                      P
o
*  *
3 *     * 2
*        *
Q o * * * * * o A
*       4      *
3  *                 * 2
*                    *
R o * * * * * * * * * * * o B
?

We have: $\Delta PBR$

$QA$ joins the midpoints of two sides.

Hence, $QA \parallel RB$ and $QA = \frac{1}{2}(RB)$

Therefore: . $RB \:=\:8$

Hi (: Thanks for your reply, it is correct.

Can you explain how you got $QA = \frac{1}{2}(RB)$? And also, does it mean that if QA joins the midpoints of two sides, QA then parallel RB? Does this rule always apply or are there limiations?

Thanks again!! (:

4. Soroban used the similarity rule.

The two triangles are similar (same proportions).

From this rule, you can easily deduce that RB = 2 * QA, since PR=2*PQ.

Hope that helps.