Math / Geometry

QuestionRefer to the figure to determine which is a true statement for the given information. $1 \overline{F G}$ is an altitude. A $\angle D G F$ is a right angle. B $D F=E F$ C $D G=G E$ D $\angle D F G \cong \angle E F G$ $2 \overline{F G}$ is a median. F $\angle D G F$ is a right angle. G $D F=E F$ H $D G=G E$ J $\angle D F G \cong \angle E F G$ 3 If $\overline{R U}$ is an altitude for $\triangle R S T$ , find $x$ .

Studdy Solution

STEP 1

What is this asking? We're looking at a triangle and figuring out what's true if a line inside is an altitude, and what's true if it's a median.
Plus, we need to solve for $x$ in another triangle with an altitude. Watch out! Don't mix up altitudes and medians!
An altitude makes a right angle, and a median splits the opposite side in half.

STEP 2

1. Altitude Analysis
2. Median Analysis
3. Altitude Calculation

STEP 3

If $\overline{FG}$ is an altitude, that means it makes a right angle with the side it hits.
Look closely!
We can see a right angle symbol where $\overline{FG}$ meets $\overline{DE}$ , specifically at point $G$ .
This means $\angle DGF$ is a right angle!

STEP 4

So, if $\overline{FG}$ is an altitude, the true statement is that $\angle DGF$ is a right angle.

STEP 5

Now, let's pretend $\overline{FG}$ is a median.
A median splits the side it hits right in half.
So, if $\overline{FG}$ is a median, it means $DG = GE$ .
The lengths of those two segments are equal!

STEP 6

We're given that $\overline{RU}$ is an altitude in $\triangle RST$ .
Altitudes form right angles, so $\angle RUS$ is a right angle, meaning it measures $90^\circ$ .

STEP 7

We're also told that $\angle RUT$ is $4x + 10$ degrees.
Since $\angle RUS$ is a right angle, we know that $\angle RUT$ and $\angle RUS$ add up to $90^\circ$ because they are complementary angles.

STEP 8

Let's write that down as an equation: $m\angle RUT + m\angle RUS = 90^\circ$ .
Substituting the given values, we get $(4x + 10) + 90 = 90$ .

STEP 9

Simplify the equation: $4x + 100 = 90$ .

STEP 10

To isolate $x$ , we'll subtract 100 from both sides: $4x + 100 - 100 = 90 - 100$ , which simplifies to $4x = -10$ .

STEP 11

Now, divide both sides by 4 to solve for $x$ : $\frac{4x}{4} = \frac{-10}{4}$ , which gives us $x = -\frac{10}{4}$ .

STEP 12

We can simplify the fraction by dividing both the numerator and the denominator by 2: $x = -\frac{5}{2}$ .

STEP 13

For the first part, if $\overline{FG}$ is an altitude, then $\angle DGF$ is a right angle (A).
If $\overline{FG}$ is a median, then $DG = GE$ (H).
For the triangle problem, $x = -\frac{5}{2}$ .

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Studdy Solution

STEP 1

STEP 2

1. Altitude Analysis
2. Median Analysis
3. Altitude Calculation

STEP 3

If $\overline{FG}$ is an altitude, that means it makes a right angle with the side it hits.
Look closely!
We can see a right angle symbol where $\overline{FG}$ meets $\overline{DE}$ , specifically at point $G$ .
This means $\angle DGF$ is a right angle!

STEP 4

So, if $\overline{FG}$ is an altitude, the true statement is that $\angle DGF$ is a right angle.

STEP 5

Now, let's pretend $\overline{FG}$ is a median.
A median splits the side it hits right in half.
So, if $\overline{FG}$ is a median, it means $DG = GE$ .
The lengths of those two segments are equal!

STEP 6

We're given that $\overline{RU}$ is an altitude in $\triangle RST$ .
Altitudes form right angles, so $\angle RUS$ is a right angle, meaning it measures $90^\circ$ .

STEP 7

We're also told that $\angle RUT$ is $4x + 10$ degrees.
Since $\angle RUS$ is a right angle, we know that $\angle RUT$ and $\angle RUS$ add up to $90^\circ$ because they are complementary angles.

STEP 8

Let's write that down as an equation: $m\angle RUT + m\angle RUS = 90^\circ$ .
Substituting the given values, we get $(4x + 10) + 90 = 90$ .

STEP 9

Simplify the equation: $4x + 100 = 90$ .

STEP 10

To isolate $x$ , we'll subtract 100 from both sides: $4x + 100 - 100 = 90 - 100$ , which simplifies to $4x = -10$ .

STEP 11

Now, divide both sides by 4 to solve for $x$ : $\frac{4x}{4} = \frac{-10}{4}$ , which gives us $x = -\frac{10}{4}$ .

STEP 12

We can simplify the fraction by dividing both the numerator and the denominator by 2: $x = -\frac{5}{2}$ .

STEP 13

For the first part, if $\overline{FG}$ is an altitude, then $\angle DGF$ is a right angle (A).
If $\overline{FG}$ is a median, then $DG = GE$ (H).
For the triangle problem, $x = -\frac{5}{2}$ .

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Studdy Solution

STEP 1

STEP 2

1. Altitude Analysis2. Median Analysis3. Altitude Calculation

STEP 3

If FG‾\overline{FG}FG is an altitude, that means it makes a right angle with the side it hits.Look closely!We can see a right angle symbol where FG‾\overline{FG}FG meets DE‾\overline{DE}DE, specifically at point GGG.This means ∠DGF\angle DGF∠DGF is a **right angle**!

STEP 4

So, if FG‾\overline{FG}FG is an altitude, the true statement is that ∠DGF\angle DGF∠DGF is a right angle.

STEP 5

Now, let's pretend FG‾\overline{FG}FG is a median.A median splits the side it hits right in half.So, if FG‾\overline{FG}FG is a median, it means DG=GEDG = GEDG=GE.The lengths of those two segments are **equal**!

STEP 6

We're given that RU‾\overline{RU}RU is an altitude in △RST\triangle RST△RST.Altitudes form right angles, so ∠RUS\angle RUS∠RUS is a right angle, meaning it measures 90∘90^\circ90∘.

STEP 7

We're also told that ∠RUT\angle RUT∠RUT is 4x+104x + 104x+10 degrees.Since ∠RUS\angle RUS∠RUS is a right angle, we know that ∠RUT\angle RUT∠RUT and ∠RUS\angle RUS∠RUS add up to 90∘90^\circ90∘ because they are complementary angles.

STEP 8

Let's write that down as an equation: m∠RUT+m∠RUS=90∘m\angle RUT + m\angle RUS = 90^\circm∠RUT+m∠RUS=90∘.Substituting the given values, we get (4x+10)+90=90(4x + 10) + 90 = 90(4x+10)+90=90.

STEP 9

**Simplify** the equation: 4x+100=904x + 100 = 904x+100=90.

STEP 10

To **isolate** xxx, we'll **subtract** 100 from both sides: 4x+100−100=90−1004x + 100 - 100 = 90 - 1004x+100−100=90−100, which simplifies to 4x=−104x = -104x=−10.

STEP 11

Now, **divide** both sides by 4 to solve for xxx: 4x4=−104\frac{4x}{4} = \frac{-10}{4}44x​=4−10​, which gives us x=−104x = -\frac{10}{4}x=−410​.

STEP 12

We can **simplify** the fraction by dividing both the numerator and the denominator by 2: x=−52x = -\frac{5}{2}x=−25​.

STEP 13

For the first part, if FG‾\overline{FG}FG is an altitude, then ∠DGF\angle DGF∠DGF is a right angle (A).If FG‾\overline{FG}FG is a median, then DG=GEDG = GEDG=GE (H).For the triangle problem, x=−52x = -\frac{5}{2}x=−25​.

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1. Altitude Analysis
2. Median Analysis
3. Altitude Calculation

If $\overline{FG}$ is an altitude, that means it makes a right angle with the side it hits.
Look closely!
We can see a right angle symbol where $\overline{FG}$ meets $\overline{DE}$ , specifically at point $G$ .
This means $\angle DGF$ is a right angle!

So, if $\overline{FG}$ is an altitude, the true statement is that $\angle DGF$ is a right angle.

Now, let's pretend $\overline{FG}$ is a median.
A median splits the side it hits right in half.
So, if $\overline{FG}$ is a median, it means $DG = GE$ .
The lengths of those two segments are equal!

We're given that $\overline{RU}$ is an altitude in $\triangle RST$ .
Altitudes form right angles, so $\angle RUS$ is a right angle, meaning it measures $90^\circ$ .

We're also told that $\angle RUT$ is $4x + 10$ degrees.
Since $\angle RUS$ is a right angle, we know that $\angle RUT$ and $\angle RUS$ add up to $90^\circ$ because they are complementary angles.

Let's write that down as an equation: $m\angle RUT + m\angle RUS = 90^\circ$ .
Substituting the given values, we get $(4x + 10) + 90 = 90$ .

Simplify the equation: $4x + 100 = 90$ .

To isolate $x$ , we'll subtract 100 from both sides: $4x + 100 - 100 = 90 - 100$ , which simplifies to $4x = -10$ .

Now, divide both sides by 4 to solve for $x$ : $\frac{4x}{4} = \frac{-10}{4}$ , which gives us $x = -\frac{10}{4}$ .

We can simplify the fraction by dividing both the numerator and the denominator by 2: $x = -\frac{5}{2}$ .

For the first part, if $\overline{FG}$ is an altitude, then $\angle DGF$ is a right angle (A).
If $\overline{FG}$ is a median, then $DG = GE$ (H).
For the triangle problem, $x = -\frac{5}{2}$ .