Math  /  Geometry

QuestionUVVW.\overline{U V} \cong \overline{V W} .
Which term describes πv\overline{\pi v} ? altitude median perpendicular bisector none of these Submit Work Not feeling ready y

Studdy Solution

STEP 1

What is this asking? If we know two sides of a triangle are equal, what do we call the line from the point connecting those sides to the opposite side? Watch out! Don't mix up the definitions of altitude, median, and perpendicular bisector!
They all involve lines inside triangles, but they mean different things.

STEP 2

1. Analyze the given information
2. Consider the definition of each term
3. Determine the correct term

STEP 3

We're told that UV\overline{UV} and VW\overline{VW} are congruent, meaning they have the *same length*.
So, UVVW\overline{UV} \cong \overline{VW}.
This makes WUV\triangle WUV an *isosceles triangle* because it has two equal sides!

STEP 4

We're also given a line segment VT\overline{VT} that goes from the vertex VV to side WU\overline{WU}.
We need to figure out what to call VT\overline{VT}.

STEP 5

An **altitude** is a line segment from a vertex that is *perpendicular* to the opposite side.
We don't know if VT\overline{VT} is perpendicular to WU\overline{WU}, so we can't say it's an altitude.

STEP 6

A **median** is a line segment from a vertex to the *midpoint* of the opposite side.
We don't know if TT is the midpoint of WU\overline{WU}, so we can't say it's a median.

STEP 7

A **perpendicular bisector** is a line segment that is both *perpendicular* to a side and goes through its *midpoint*.
Again, we don't have enough information to say this about VT\overline{VT}.

STEP 8

However, since WUV\triangle WUV is an *isosceles triangle*, and VT\overline{VT} extends from the vertex connecting the two equal sides, it *must* be an **altitude**, a **median**, and a **perpendicular bisector**!
This is a special property of isosceles triangles.

STEP 9

So, VT\overline{VT} is perpendicular to WU\overline{WU}, meaning it's an **altitude**.
It also bisects WU\overline{WU}, meaning TT is the midpoint of WU\overline{WU}, making VT\overline{VT} a **median** and a **perpendicular bisector**.

STEP 10

VT\overline{VT} is an altitude, a median, *and* a perpendicular bisector!

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