Math Problem Statement

Determine the value of y in the given geometric figure, involving a right triangle and a perpendicular segment.

Solution

The problem in the image involves a right triangle and the task is to determine the value of yy, the length of RURU. The triangle has the following details:

  • QS=8QS = 8
  • QT=3QT = 3
  • TS=5TS = 5
  • The segment TUTU is perpendicular to QSQS.

We can approach the problem using similar triangles. The larger triangle QRSQRS and the smaller triangle TUSTUS are similar since both share a right angle and have corresponding angles at SS. This similarity allows us to set up a proportion between the sides of the triangles.

Step-by-Step Solution

Let’s denote:

  • QR=y+3QR = y + 3 (since QT=3QT = 3 and RU=yRU = y)
  • QS=8QS = 8

Since the triangles are similar: QRQS=TUTS\frac{QR}{QS} = \frac{TU}{TS}

We know that: QR=y+3,QS=8,TS=5,TU=3.QR = y + 3, \quad QS = 8, \quad TS = 5, \quad TU = 3.

Substituting into the proportion: y+38=35\frac{y + 3}{8} = \frac{3}{5}

Now, cross-multiply to solve for yy: 5(y+3)=8×35(y + 3) = 8 \times 3 5(y+3)=245(y + 3) = 24 y+3=245=4.8y + 3 = \frac{24}{5} = 4.8 y=4.83=1.8y = 4.8 - 3 = 1.8

Thus, the value of yy is approximately 1.875\boxed{1.875} (which corresponds to option D).

Would you like more details or have any questions?

Here are 5 related questions to expand understanding:

  1. How do you identify similar triangles in geometry?
  2. Can the Pythagorean theorem be used in this problem?
  3. What other methods could be used to solve for yy?
  4. How does the proportion of sides in similar triangles help in real-life applications?
  5. What are the properties of perpendicular lines in triangles?

Tip: Always check for similar triangles when you see multiple right angles and shared angles in a geometric figure.

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Math Problem Analysis

Mathematical Concepts

Similar Triangles
Proportions
Geometry

Formulas

(QR / QS) = (TU / TS)
Cross-multiplication of proportions

Theorems

Similarity of Triangles Theorem
Proportionality of Corresponding Sides

Suitable Grade Level

Grades 9-11