F4 Nature of roots

Q1.
On a football pitch , a boy runs due east frompoint P to Q, and then runs due north to point R. The total distance he hastravelled is 92 m . If PR = 68 m, find the possible distances between P and Q.

Let PQ = y m

Totaldistance he has travelled :
PQ + QR = 92 m
(y m) + QR = 92 m
QR = (92 - y) m

ΔPQR is a right-angled triangle with ÐQ = 90°
In ΔPQR : PQ² + QR² = PR² (Pythagoreantheorem)
y² + (92 - y)² = 68²
y² + 8464 - 184y + y² = 4624
2y² - 184y + 3840 = 0
y² - 92y + 1920 = 0
(y - 60)(y - 32) = 0
y = 60 or y = 32

Hence, PQ is 60 m or 32 m.


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Q2.
Abook AB leans aginst a bookshelf . It is given that AB = 20 cm and BO = 12 cm . If A is lowered by x cm to A' so that Bslides the distances to B' , find the value of x .

In ΔABO :
AO² + BO² = AB²
AO² + (12 cm)² = (20 cm)²
AO² = 256 cm²
AO = 16 cm

Refer to the sentence " If A is lowered by x cm to A' so that B slidesthe distances to B'".
Some information on B' or on ΔA'B'O is missed.
There is insufficient information to calculate x.