Math Problem Statement

Triangle S R Q is shown. Angle S R Q is a right angle. An altitude is drawn from point R to point T on side S Q to form a right angle. The length of S T is 9, the length of T Q is 16, and the length of R Q is x. What is the value of x?

12 units 15 units 20 units 24 units

Solution

We have a right triangle SRQ\triangle SRQ with an altitude RTRT drawn from RR to TT on SQSQ, forming another right angle. Given:

  • ST=9ST = 9
  • TQ=16TQ = 16
  • RQ=xRQ = x

Step 1: Find SQSQ

Since STST and TQTQ are segments of SQSQ:

SQ=ST+TQ=9+16=25SQ = ST + TQ = 9 + 16 = 25

Step 2: Use the Geometric Mean Theorem

The altitude theorem states:

RT2=STTQRT^2 = ST \cdot TQ

RT2=9×16=144RT^2 = 9 \times 16 = 144

RT=144=12RT = \sqrt{144} = 12

Step 3: Use the Pythagorean Theorem in RQT\triangle RQ T

Since RTQ\angle RTQ is a right angle:

RQ2=RT2+TQ2RQ^2 = RT^2 + TQ^2

x2=122+162x^2 = 12^2 + 16^2

x2=144+256=400x^2 = 144 + 256 = 400

x=400=20x = \sqrt{400} = 20

Answer:

20 units\boxed{20} \text{ units}

Would you like a detailed breakdown of any step?

Related Questions:

  1. What is the altitude theorem in right triangles, and how does it apply here?
  2. Can you find the length of SRSR in this triangle?
  3. What is the Pythagorean Theorem, and how was it used to find xx?
  4. If RTRT was given instead of needing to calculate it, how would that change the approach?
  5. How can we use similarity of triangles to verify the result?

Tip:

The altitude theorem states that the altitude to the hypotenuse in a right triangle creates two smaller right triangles that are similar to the original triangle.

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Math Problem Analysis

Mathematical Concepts

Right Triangle Geometry
Altitude Theorem
Pythagorean Theorem

Formulas

SQ = ST + TQ
RT^2 = ST * TQ
RQ^2 = RT^2 + TQ^2

Theorems

Geometric Mean Theorem (Altitude Theorem)
Pythagorean Theorem

Suitable Grade Level

Grades 9-11