Math Problem Statement

solve it

Solve for x, AD, and CD in the given geometric diagram where BD = 2x + 10 and AD = 3x + 2, with BD = CD.

Solution

In the given geometric diagram, we have a triangle ABC\triangle ABC, where ADAD is perpendicular to BCBC, creating a right triangle at DD. The expressions for side lengths are given as:

  • BD=2x+10BD = 2x + 10
  • DA=3x+2DA = 3x + 2

We can assume that ABD\triangle ABD and CAD\triangle CAD are congruent by the right angle and the corresponding equal sides, implying that BD=CDBD = CD. Let's solve for xx and the side lengths.

Step 1: Set up the equation for BD=CDBD = CD

Since ABDACD\triangle ABD \cong \triangle ACD, we have: BD=CDBD = CD Thus: 2x+10=3x+22x + 10 = 3x + 2

Step 2: Solve for xx

We subtract 2x2x from both sides: 10=x+210 = x + 2 Now, subtract 2 from both sides: x=8x = 8

Step 3: Find the lengths of ADAD and CDCD

Now that we have x=8x = 8, substitute into the expressions for ADAD and CDCD:

  • AD=3x+2=3(8)+2=24+2=26AD = 3x + 2 = 3(8) + 2 = 24 + 2 = 26
  • BD=2x+10=2(8)+10=16+10=26BD = 2x + 10 = 2(8) + 10 = 16 + 10 = 26

Since BD=CDBD = CD, we conclude that: CD=26CD = 26

Final answers:

  • x=8x = 8
  • AD=26AD = 26
  • CD=26CD = 26

Would you like further details or have any other questions? Here are 5 related questions to explore further:

  1. What are the criteria for triangles to be congruent?
  2. How do you solve for unknowns in geometric problems with algebraic expressions?
  3. What is the Pythagorean theorem, and can it be applied here?
  4. How do perpendicular bisectors relate to triangle properties?
  5. What are the properties of an isosceles triangle?

Tip: In geometry, identifying congruent triangles can often simplify calculations by relating side lengths directly through equality.

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Math Problem Analysis

Mathematical Concepts

Algebra
Congruent Triangles
Right Triangles

Formulas

BD = CD
Algebraic equation solving

Theorems

Congruent Triangles Theorem
Properties of Isosceles Triangles

Suitable Grade Level

Grades 8-10