### Solving for $ x $ in an Isosceles Triangle**Problem Statement:**Given that $ AB = 8x + 5 $, $ CB = 10x - 1 $, and $ BD \perp AC $, solve for $ x $. A triangle $ ABC $ is illustrated where $ BD $ is the altitude from $ B $ to $ AC $, making $\angle BDA$ and $\angle BDC$ right angles. **Steps to Solve:**1. **Understanding the Isosceles Triangle:** Since $ AB $ and $ CB $ are given by two distinct expressions but the problem involves isosceles properties (indicated by mid-perpendicular lines indicating equality), we can set $ AB $ equal to $ CB $.2. **Set the expressions equal:** \[ 8x + 5 = 10x - 1 \]3. **Isolate $ x $:** \[ 10x - 8x = 5 + 1 \] \[ 2x = 6 \] \[ x = 3 \]4. **Verify the Value:** - Plug $ x = 3 $ back into the expressions to ensure both sides are equal: - For $ AB $: $ 8(3) + 5 = 24 + 5 = 29 $ - For $ CB $: $ 10(3) - 1 = 30 - 1 = 29 $Both sides equal $ 29 $, confirming the solution is correct. 5. **Conclusion:** Therefore, the value of $ x $ is: \[ x = 3 \]**Visual Explanation:**- **Diagram Description:** The diagram shows triangle $ ABC $ with $ BD $ as the altitude. The marks on $ AB $ and $ CB $ indicate these two sides are equal in length (as it is an isosceles triangle with $ AB = CB $). By following the steps above, students can solve for the variable $ x $ in isosceles triangles given similar length expressions. The diagram reinforces the understanding of altitude and properties specific to isosceles triangles. ### Example Question from Geometry: Understanding Isosceles Triangles#### Question:In the figure given, triangle ABC is isosceles with $ AB = AC $. Point D is the midpoint of the base BC, and the segment AD is perpendicular to BC.Given that:- Triangle ABC is an isosceles triangle with AB = AC- Point D is the midpoint of BC- Segment AD is perpendicular to BCWhat is the ratio of the lengths $ \frac{AD}{DB} $ (denoted as $ x $)?#### Choose the correct answer from the options below:- $ x = 2 $- $ x = 1.5 $- $ x = 3 $- $ x = 1 $#### Explanation:In an isosceles triangle, the altitude from the vertex angle bisects the base into two equal parts. Since $ D $ is the midpoint of $ BC $, and $ AD $ is perpendicular to $ BC $, the triangles $ ABD $ and $ ACD $ are congruent.Let's explore the possible values for $ x $ based on the information given in the problem. The correct ratio can be determined using properties of isosceles triangles and perpendicular bisectors.### Answer:Use the properties of the isosceles triangle to solve this problem. The correct ratio $ x $ can be deduced by understanding the nature of the triangles and applying the appropriate theorems or formulas.---This explanation ensures that students understand the question and can apply geometric properties to find the right answer.

Answered: Image /qna-images/answer/39552ddf-ccfe-45be-a73c-65bc997ea3f4.jpg

Answered: 3. If AB = 8x + 5, CB = 10x - 1, and BD…

3. If AB = 8x + 5, CB = 10x - 1, and BD AC, 1 solve for x.

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