Math Problem Statement

Solve for x in a geometry problem involving two similar triangles where AB = x, BC = 1, BD = 9, and ED = 5.

Solution

The image you uploaded shows a right triangle with several labeled sides and a problem asking to solve for xx. Here’s how we can approach this problem step by step:

Given:

  • Triangle ABC\triangle ABC and a larger triangle ADE\triangle ADE are drawn.
  • We know that:
    • AB=xAB = x
    • BC=1BC = 1 (height of the smaller triangle)
    • BD=9BD = 9 (total base from BB to DD)
    • ED=5ED = 5 (height of the larger triangle)

This seems like a case of similar triangles because both triangles share angle A\angle A, and both have right angles at BB and DD. Therefore, triangles ABC\triangle ABC and ADE\triangle ADE are similar by AA similarity.

Step-by-step solution:

  1. Set up the similarity ratio: Since the triangles are similar, the corresponding sides are proportional: ABAD=BCED\frac{AB}{AD} = \frac{BC}{ED} Substituting the known values: xx+9=15\frac{x}{x + 9} = \frac{1}{5}
  2. Solve for xx: Cross-multiply to eliminate the fractions: 5x=(x+9)5x = (x + 9) Simplifying this equation: 5x=x+95x = x + 9 Subtract xx from both sides: 4x=94x = 9 Finally, divide both sides by 4: x=94=2.25x = \frac{9}{4} = 2.25

Thus, the value of xx is 2.25.


Let me know if you'd like more details on any part of this solution!

5 Related Questions:

  1. How do you prove that two triangles are similar?
  2. What is the AA similarity theorem in geometry?
  3. How do you find the length of a missing side using proportions?
  4. Can you use trigonometric ratios to solve for unknown sides in similar triangles?
  5. How do you solve equations involving cross-multiplication?

Tip:

Always check for shared angles and corresponding right angles in geometry problems. This can quickly help identify similar triangles and lead to solving for unknown sides using proportions.

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Math Problem Analysis

Mathematical Concepts

Geometry
Similar Triangles
Proportions
Cross Multiplication

Formulas

\(\frac{AB}{AD} = \frac{BC}{ED}\)
Cross-multiplication formula

Theorems

AA Similarity Theorem

Suitable Grade Level

Grades 8-10