Math / Geometry

QuestionIf $P$ is the orthocenter of $\triangle ABC$ , $AB = 13$ , $BF = 9$ , and $FC = 5.6$ , find the perimeter of $\triangle ABC$ .

Studdy Solution

STEP 1

1. $P$ is the orthocenter of $\triangle ABC$ .
2. $AB = 13$ , $BF = 9$ , and $FC = 5.6$ .
3. We need to find the perimeter of $\triangle ABC$ .

STEP 2

1. Understand the properties of the orthocenter and the given triangle.
2. Use the given segment lengths to find the length of side $AC$ .
3. Calculate the length of side $BC$ .
4. Sum the lengths of $AB$ , $BC$ , and $AC$ to find the perimeter.

STEP 3

The orthocenter $P$ of a triangle is the point where the three altitudes intersect. The altitudes are perpendicular to the opposite sides. In $\triangle ABC$ , $P$ does not directly help in finding the perimeter, but it confirms the triangle's properties.

STEP 4

We are given $BF = 9$ and $FC = 5.6$ . Since $F$ is on $\overline{AC}$ , the length of $AC$ is the sum of $BF$ and $FC$ :
$AC = BF + FC = 9 + 5.6 = 14.6$

STEP 5

To find $BC$ , we use the triangle inequality or other geometric properties, but since $AB = 13$ , $AC = 14.6$ , and $BF = 9$ , $BC$ can be calculated by considering the triangle's properties and given segments.

STEP 6

Since $F$ is on $AC$ and $BF = 9$ , $BC$ is the remaining segment on $AC$ :
$BC = AC - FC = 14.6 - 5.6 = 9$

STEP 7

Now, calculate the perimeter of $\triangle ABC$ by summing the lengths of its sides:
$\text{Perimeter} = AB + BC + AC$ $= 13 + 9 + 14.6$ $= 36.6$
The perimeter of $\triangle ABC$ is:
$\boxed{36.6}$

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QuestionIf $P$ is the orthocenter of $\triangle ABC$ , $AB = 13$ , $BF = 9$ , and $FC = 5.6$ , find the perimeter of $\triangle ABC$ .

Studdy Solution

STEP 1

1. $P$ is the orthocenter of $\triangle ABC$ .
2. $AB = 13$ , $BF = 9$ , and $FC = 5.6$ .
3. We need to find the perimeter of $\triangle ABC$ .

STEP 2

1. Understand the properties of the orthocenter and the given triangle.
2. Use the given segment lengths to find the length of side $AC$ .
3. Calculate the length of side $BC$ .
4. Sum the lengths of $AB$ , $BC$ , and $AC$ to find the perimeter.

STEP 3

The orthocenter $P$ of a triangle is the point where the three altitudes intersect. The altitudes are perpendicular to the opposite sides. In $\triangle ABC$ , $P$ does not directly help in finding the perimeter, but it confirms the triangle's properties.

STEP 4

We are given $BF = 9$ and $FC = 5.6$ . Since $F$ is on $\overline{AC}$ , the length of $AC$ is the sum of $BF$ and $FC$ :
$AC = BF + FC = 9 + 5.6 = 14.6$

STEP 5

To find $BC$ , we use the triangle inequality or other geometric properties, but since $AB = 13$ , $AC = 14.6$ , and $BF = 9$ , $BC$ can be calculated by considering the triangle's properties and given segments.

STEP 6

Since $F$ is on $AC$ and $BF = 9$ , $BC$ is the remaining segment on $AC$ :
$BC = AC - FC = 14.6 - 5.6 = 9$

STEP 7

Now, calculate the perimeter of $\triangle ABC$ by summing the lengths of its sides:
$\text{Perimeter} = AB + BC + AC$ $= 13 + 9 + 14.6$ $= 36.6$
The perimeter of $\triangle ABC$ is:
$\boxed{36.6}$

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QuestionIf P P P is the orthocenter of △ABC\triangle ABC△ABC, AB=13 AB = 13 AB=13, BF=9 BF = 9 BF=9, and FC=5.6 FC = 5.6 FC=5.6, find the perimeter of △ABC\triangle ABC△ABC.

Studdy Solution

STEP 1

1. P P P is the orthocenter of △ABC\triangle ABC△ABC.2. AB=13 AB = 13 AB=13, BF=9 BF = 9 BF=9, and FC=5.6 FC = 5.6 FC=5.6.3. We need to find the perimeter of △ABC\triangle ABC△ABC.

STEP 2

1. Understand the properties of the orthocenter and the given triangle.2. Use the given segment lengths to find the length of side AC AC AC.3. Calculate the length of side BC BC BC.4. Sum the lengths of AB AB AB, BC BC BC, and AC AC AC to find the perimeter.

STEP 3

The orthocenter P P P of a triangle is the point where the three altitudes intersect. The altitudes are perpendicular to the opposite sides. In △ABC\triangle ABC△ABC, P P P does not directly help in finding the perimeter, but it confirms the triangle's properties.

STEP 4

We are given BF=9 BF = 9 BF=9 and FC=5.6 FC = 5.6 FC=5.6. Since F F F is on AC‾ \overline{AC} AC, the length of AC AC AC is the sum of BF BF BF and FC FC FC:AC=BF+FC=9+5.6=14.6 AC = BF + FC = 9 + 5.6 = 14.6 AC=BF+FC=9+5.6=14.6

STEP 5

To find BC BC BC, we use the triangle inequality or other geometric properties, but since AB=13 AB = 13 AB=13, AC=14.6 AC = 14.6 AC=14.6, and BF=9 BF = 9 BF=9, BC BC BC can be calculated by considering the triangle's properties and given segments.

STEP 6

Since F F F is on AC AC AC and BF=9 BF = 9 BF=9, BC BC BC is the remaining segment on AC AC AC:BC=AC−FC=14.6−5.6=9 BC = AC - FC = 14.6 - 5.6 = 9 BC=AC−FC=14.6−5.6=9

STEP 7

Now, calculate the perimeter of △ABC\triangle ABC△ABC by summing the lengths of its sides:Perimeter=AB+BC+AC \text{Perimeter} = AB + BC + AC Perimeter=AB+BC+AC =13+9+14.6 = 13 + 9 + 14.6 =13+9+14.6 =36.6 = 36.6 =36.6 The perimeter of △ABC\triangle ABC△ABC is:36.6 \boxed{36.6} 36.6​

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QuestionIf $P$ is the orthocenter of $\triangle ABC$ , $AB = 13$ , $BF = 9$ , and $FC = 5.6$ , find the perimeter of $\triangle ABC$ .

1. $P$ is the orthocenter of $\triangle ABC$ .
2. $AB = 13$ , $BF = 9$ , and $FC = 5.6$ .
3. We need to find the perimeter of $\triangle ABC$ .

1. Understand the properties of the orthocenter and the given triangle.
2. Use the given segment lengths to find the length of side $AC$ .
3. Calculate the length of side $BC$ .
4. Sum the lengths of $AB$ , $BC$ , and $AC$ to find the perimeter.

The orthocenter $P$ of a triangle is the point where the three altitudes intersect. The altitudes are perpendicular to the opposite sides. In $\triangle ABC$ , $P$ does not directly help in finding the perimeter, but it confirms the triangle's properties.

We are given $BF = 9$ and $FC = 5.6$ . Since $F$ is on $\overline{AC}$ , the length of $AC$ is the sum of $BF$ and $FC$ :
$AC = BF + FC = 9 + 5.6 = 14.6$

To find $BC$ , we use the triangle inequality or other geometric properties, but since $AB = 13$ , $AC = 14.6$ , and $BF = 9$ , $BC$ can be calculated by considering the triangle's properties and given segments.

Since $F$ is on $AC$ and $BF = 9$ , $BC$ is the remaining segment on $AC$ :
$BC = AC - FC = 14.6 - 5.6 = 9$

Now, calculate the perimeter of $\triangle ABC$ by summing the lengths of its sides:
$\text{Perimeter} = AB + BC + AC$ $= 13 + 9 + 14.6$ $= 36.6$
The perimeter of $\triangle ABC$ is:
$\boxed{36.6}$