Question

similar right triangles 1. identify the similar triangles in the diagram below by writing a similarity statement. then, sketch and label the triangles separately so they have the same orientation. 2. find the value of x. 3.

Explanation:

Step1: (For Q1) Identify similar triangles

The large right triangle $\triangle STU$ (right angle at $T$), the smaller right triangle $\triangle STV$ (right angle at $V$), and $\triangle TUV$ (right angle at $V$) are similar. The similarity statement is $\triangle STU \sim \triangle TVU \sim \triangle STV$. When sketched with matching orientation:

1. $\triangle STU$: right angle at $T$, sides $ST$, $TU$, $SU$
2. $\triangle TVU$: right angle at $V$, sides $TV$, $VU$, $TU$
3. $\triangle STV$: right angle at $V$, sides $ST$, $TV$, $SV$

Step2: (For Q2) Use geometric mean theorem

For a right triangle, the altitude to hypotenuse is $\frac{\text{product of legs}}{\text{hypotenuse}}$.
$\text{Legs}=16, 30$, $\text{Hypotenuse}=34$
$x = \frac{16 \times 30}{34}$
Simplify: $x = \frac{480}{34} = \frac{240}{17}$

Step3: (For Q3) Use geometric mean theorem

First find hypotenuse of large triangle:
$\text{Hypotenuse} = \sqrt{20^2 + 21^2} = \sqrt{400 + 441} = \sqrt{841} = 29$
Altitude $x = \frac{20 \times 21}{29}$
Calculate: $x = \frac{420}{29}$

Answer:

1. Similarity statement: $\triangle STU \sim \triangle TVU \sim \triangle STV$. Sketches:

• Triangle 1: Right angle at top, vertices labeled $S$, $T$, $U$ (right at $T$)
• Triangle 2: Right angle at bottom, vertices labeled $T$, $V$, $U$ (right at $V$)
• Triangle 3: Right angle at bottom, vertices labeled $S$, $T$, $V$ (right at $V$)

1. $\frac{240}{17}$
2. $\frac{420}{29}$