Question

ABCD is a trapezium. P is a point along AC such that AP=4PC. DC=1/4AB.

a) express PB in terms of a and b in its simplest form

b) express DP in terms of a and b in its simplest form

c) does DPB form a straight line?

Answer 1

To find the value of PB in terms of a and b, we need to use the fact that AP is 4 times PC. By expressing the lengths of the sides in terms of a, we can simplify the equation and find that PB is equal to a + b.

Step-by-step explanation:

To find the value of PB, we need to use the fact that AP is 4 times PC. Let's assign a value to one of the sides of the trapezium. Let's say AB is equal to 'a' and DC is equal to 'b'.

Since DC is equal to 1/4 of AB, we can write: b = (1/4)a

Since P divides AC into a 4:1 ratio, we can write: AP = 4PC

Let's express the lengths of the sides in terms of 'a'.

AC = AB + BC = a + b

PC = (1/5) AC = (1/5)(a + b)

AP = 4PC = 4(1/5)(a + b) = 4/5(a + b)

From the given information, we know that AP is equal to 4 times PC. Therefore,

4/5(a + b) = 4(1/5)(a + b)

Simplifying this equation gives us 4/5(a + b) = 4/5(a + b). This equation is true for any values of a and b, so we can say that PB is equal to a + b.

Therefore, PB = a + b.

Answer 2

a) We can use similar triangles to find PB in terms of a and b. Let x be the length of AD. Then, using the fact that AP = 4PC, we have:

PC = CP = x - b

AP = 4(x - b)

Also, using the fact that DC = (1/4)AB, we have:

AD = x

AB = 4DC = x/4

BC = AB - AD = x/4 - x = -3x/4

Now, consider the similar triangles PBC and ABD:

PB/AB = BC/AD

PB/(x/4) = (-3x/4)/x

PB = -3/4(x/4) = -3x/16

Finally, substituting x = a + b, we have:

PB = -3(a + b)/16

b) Using the same similar triangles as in part (a), we have:

DP/DC = PB/BC

DP/(1/4)AB = PB/(-3x/4)

DP = -3/4(PB)(DC/BC)AB

DP = -3/4(PB)(1/4)/(AB - AD)AB

Substituting the expressions for PB, AB, and AD from part (a), we get:

DP = -3(a + b)/16 * 1/4 / (-3(a + b)/4) * (a + b)/4

DP = -3/16 * 1/4 * 4/(3(a + b)) * (a + b)

DP = -3/16

So, DP = -3/16(a + b)

c) To check if DPB forms a straight line, we need to verify if the slopes of DP and PB are equal. Using the expressions we found in parts (a) and (b), we have:

Slope of PB = Δy/Δx = (-3/16(a+b) - 0)/(0 - (-3(a+b)/16)) = 3/16

Slope of DP = Δy/Δx = (-3(a+b)/16 - (-3/16(a+b)))/(1/4 - 0) = -3(a+b)/4

Since the slopes are not equal, DPB does not form a straight line.