If P is the circumcenter △ A B C, and AD =3x - 11, DB =5x - 29, PC =18, find DP.

Answer:

8.2

Step-by-step explanation:

1. Since AD is equal to DB, we will need to set 3x-11 equal to 5x-29. When you solve the equation you will get 9 for x.

2. Plug the 9 in for x in DB (you could plug it in for AD instead if you want since DB and AD are equal to each other). You should get 16.

3. Take a look at the smaller triangle DBP which is part of triangle ABC. We will need to use the Pythagorean Theorem. Square the legs of the triangle and set it equal to the hypotenuse squared. One of the legs, DP, is unknown so we will just have it as x. The other leg is DB, which is 16.

* PB is the hypotenuse and has a value of 18. Remember that P is the circumcenter, which means it is equidistant to each vertex and each angle bisector is equal to each other. As a result PC is equal to PB which happens to be 18.

You should have it something like this:

16^2 + x^2 = 18^2

Now do all the squaring and stuff and you should eventually get x^2 = 68

4. Square root x^2 = 68 and you will get a long decimal. You will need to round it to the tenths place.

5. After rounding you will finally get 8.2

The measurement of DP is 8.2 units.

The circumcenter of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect.

Given that, P is the circumcenter △ ABC, and AD = 3x - 11, DB =5x - 29,

PC = 18

Since, DP bisects AB, so, AB = DP

3x-11 = 5x-29

x = 9

DB = 5*9 - 29 = 16

Since, P is the circumcenter, then PB = PC = PA = 18

Using Pythagoras theorem in Δ BPD

PB² = DB² + DP²

DP² = PB² - DB²

DP² = 18² - 16²

DP² = 324 - 256

DP² = 68

DP = 8.2

Hence, The measurement of DP is 8.2 units.

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