Math Problem Statement

What's x

What's x?

Solution

The image shows two right triangles, STU\triangle STU and ABC\triangle ABC, and the problem states that these triangles are similar: STUABC\triangle STU \sim \triangle ABC.

Step 1: Use Similarity of Triangles

Since the triangles are similar, the corresponding sides of the triangles are proportional: STAB=TUBC=SUAC\frac{ST}{AB} = \frac{TU}{BC} = \frac{SU}{AC} In the given triangles:

  • ST=6ST = 6, TU=3TU = 3, and SUSU is the hypotenuse of triangle STU\triangle STU, which can be calculated using the Pythagorean theorem.
  • AB=36AB = 36, BC=2x6BC = 2x - 6, and AC=24AC = 24.

Step 2: Find the Length of SUSU

Apply the Pythagorean theorem in STU\triangle STU: SU2=ST2+TU2=62+32=36+9=45SU^2 = ST^2 + TU^2 = 6^2 + 3^2 = 36 + 9 = 45 SU=45=35SU = \sqrt{45} = 3\sqrt{5}

Step 3: Set up Proportions

Now use the proportion between corresponding sides: STAB=TUBC=SUAC\frac{ST}{AB} = \frac{TU}{BC} = \frac{SU}{AC} Starting with TUBC\frac{TU}{BC}: 32x6=636\frac{3}{2x - 6} = \frac{6}{36} Simplify 636\frac{6}{36} to 16\frac{1}{6}, so: 32x6=16\frac{3}{2x - 6} = \frac{1}{6}

Step 4: Solve for xx

Cross-multiply to solve for xx: 3×6=(2x6)×13 \times 6 = (2x - 6) \times 1 18=2x618 = 2x - 6 Add 6 to both sides: 24=2x24 = 2x Divide by 2: x=12x = 12

Final Answer:

The value of xx is 1212.

Would you like any further details or have any questions?

Here are 5 related questions to explore:

  1. How do you verify that two triangles are similar?
  2. What is the importance of proportionality in similar triangles?
  3. Can the Pythagorean theorem be applied in non-right triangles?
  4. What would happen if the triangles weren't similar but still shared some side ratios?
  5. How does changing one side affect the corresponding sides in similar triangles?

Tip: Always check that your proportionality ratios match between all corresponding sides in similar triangles!

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Math Problem Analysis

Mathematical Concepts

Triangle Similarity
Proportions
Pythagorean Theorem
Algebra

Formulas

Proportions: \( \frac{ST}{AB} = \frac{TU}{BC} = \frac{SU}{AC} \)
Pythagorean theorem: \( a^2 + b^2 = c^2 \)

Theorems

Similarity of Triangles
Pythagorean Theorem

Suitable Grade Level

Grades 9-10