Given the diagram below, if $ P $ is the circumcenter of $\triangle ABC$, $ AD = 8x + 3 $, $ DB = 17x - 15 $, and $ DP = 12 $, find $ PB $. Round answers to the nearest tenth when needed.**Diagram Explanation:**The diagram shows a triangle $ \triangle ABC $ with point $ P $ as its circumcenter. The lines $ AD $, $ DB $, and $ DP $ are indicated as segments within the triangle. The circumcenter $ P $ is equidistant from the vertices of the triangle, making $ PA = PB = PC $. The task is to find the length of $ PB $ given the algebraic expressions for $ AD $ and $ DB $ and the specific length of $ DP $.**Solution Steps:**1. Set $ AD = DB $ because both distances add up to the same line $ AB $.2. Solve for $ x $ using the equations: \[ 8x + 3 = 17x - 15 \]3. Simplify the equation: \[ 8x + 3 = 17x - 15 \implies 18 = 9x \implies x = 2 \]4. Substitute $ x = 2 $ back to find $ AD $: \[ AD = 8(2) + 3 = 19 \]5. Verify $ DB $ with $ x = 2 $: \[ DB = 17(2) - 15 = 19 \]6. Calculate $ PB $: - Since $ P $ is the circumcenter, and given $ DP = 12 $, $ PB = PD = 12 $.So, $ PB = 12 $.

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Answered: Given the diagram below, if P is the…

Given the diagram below, if P is the circumcenter of AABC, AD=8x+3, DB=17x- 15, and DP=12, find PB. Round answers to the nearest tenth when needed.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Concept explainers

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

Given the diagram below, if $ P $ is the circumcenter of $\triangle ABC$, $ AD = 8x + 3 $, $ DB = 17x - 15 $, and $ DP = 12 $, find $ PB $. Round answers to the nearest tenth when needed.

**Diagram Explanation:**

The diagram shows a triangle $ \triangle ABC $ with point $ P $ as its circumcenter. The lines $ AD $, $ DB $, and $ DP $ are indicated as segments within the triangle. The circumcenter $ P $ is equidistant from the vertices of the triangle, making $ PA = PB = PC $. The task is to find the length of $ PB $ given the algebraic expressions for $ AD $ and $ DB $ and the specific length of $ DP $.

**Solution Steps:**

1. Set $ AD = DB $ because both distances add up to the same line $ AB $.
2. Solve for $ x $ using the equations:
\[
8x + 3 = 17x - 15
\]
3. Simplify the equation:
\[
8x + 3 = 17x - 15 \implies 18 = 9x \implies x = 2
\]
4. Substitute $ x = 2 $ back to find $ AD $:
\[
AD = 8(2) + 3 = 19
\]
5. Verify $ DB $ with $ x = 2 $:
\[
DB = 17(2) - 15 = 19
\]
6. Calculate $ PB $:
- Since $ P $ is the circumcenter, and given $ DP = 12 $, $ PB = PD = 12 $.

So, $ PB = 12 $.

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