Question

Oints Q and R are midpoints of the sides of triangle ABC.

Triangle A B C is cut by line segment Q R. Point Q is the midpoint of side A B and point R is the midpoint of side A C. The lengths of A Q and Q B are 4 p, the length of Q R is 2 p + 3, and the length of C B is 6 p minus 4. The lengths of A R and R C are congruent.

What is AQ?

10 units
14 units
20 units
32 units

Answer 1

AQ = 20 units

Explanation:

Comparing triangle AQR to ABC,

`(AQ)/(AB)` =
`(QR)/(BC)`

`(4p)/(8p)` =
`(2p + 3)/(6p-4)`

cross multiply and make p the subject of formula, we have:

8p (2p+3) = 4p(6p-4)

16
`p^(2)` + 24p = 24
`p^(2)` - 16p

24p + 16p = 24
`p^(2)` - 16
`p^(2)`

40p = 8
`p^(2)`

divide through by 8p,

p = 5

Therefore, AQ = 4p

= 4 × 5

= 20

AQ = 20 units