# Equilateral triangle ABC

In the equilateral triangle ABC, K is the center of the AB side, the L point lies on one-third of the BC side near the point C, and the point M lies in the one-third of the side of the AC side closer to the point A. Find what part of the ABC triangle contains the triangle KLM.

Correct result:

x =  0.278

#### Solution:

$a=1 \ \\ h=\sqrt{ a^2 - (a/2)^2 }=\sqrt{ 1^2 - (1/2)^2 } \doteq 0.866 \ \\ \ \\ S=\dfrac{ a \cdot \ h }{ 2 }=\dfrac{ 1 \cdot \ 0.866 }{ 2 } \doteq 0.433 \ \\ \ \\ S_{1}=\dfrac{ \dfrac{ h }{ 3 } \cdot \ \dfrac{ a }{ 2 } }{ 2 }=\dfrac{ \dfrac{ 0.866 }{ 3 } \cdot \ \dfrac{ 1 }{ 2 } }{ 2 } \doteq 0.0722 \ \\ S_{2}=\dfrac{ \dfrac{ 2 \cdot \ h }{ 3 } \cdot \ \dfrac{ a }{ 2 } }{ 2 }=\dfrac{ \dfrac{ 2 \cdot \ 0.866 }{ 3 } \cdot \ \dfrac{ 1 }{ 2 } }{ 2 } \doteq 0.1443 \ \\ S_{3}=\dfrac{ \dfrac{ h }{ 3 } \cdot \ \dfrac{ 2 \cdot \ a }{ 3 } }{ 2 }=\dfrac{ \dfrac{ 0.866 }{ 3 } \cdot \ \dfrac{ 2 \cdot \ 1 }{ 3 } }{ 2 } \doteq 0.0962 \ \\ \ \\ S_{4}=S - (S_{1}+S_{2}+S_{3})=0.433 - (0.0722+0.1443+0.0962) \doteq 0.1203 \ \\ \ \\ x=\dfrac{ S_{4} }{ S }=\dfrac{ 0.1203 }{ 0.433 }=\dfrac{ 5 }{ 18 }=0.278$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Please write to us with your comment on the math problem or ask something. Thank you for helping each other - students, teachers, parents, and problem authors.

Tips to related online calculators
Need help calculate sum, simplify or multiply fractions? Try our fraction calculator.
Pythagorean theorem is the base for the right triangle calculator.

#### You need to know the following knowledge to solve this word math problem:

We encourage you to watch this tutorial video on this math problem:

## Next similar math problems:

• Two chords
Calculate the length of chord AB and perpendicular chord BC to circle if AB is 4 cm from the center of the circle and BC 8 cm from the center of the circle.
• Similarity coefficient
In the triangle TMA the length of the sides is t = 5cm, m = 3.5cm, a = 6.2cm. Another similar triangle has side lengths of 6.65 cm, 11.78 cm, 9.5 cm. Determine the similarity coefficient of these triangles and assign similar sides to each other.
• Similarity of two triangles
The KLM triangle has a side length of k = 6.3cm, l = 8.1cm, m = 11.1cm. The triangle XYZ has a side length of x = 8.4cm, y = 10.8cm, z = 14.8cm. Are triangle KLM and XYZ similar? (write 0 if not, if yes, find and write the coefficient of a similarity)
• Secret treasure
Scouts have a tent in the shape of a regular quadrilateral pyramid with a side of the base 4 m and a height of 3 m. Determine the radius r (and height h) of the container so that they can hide the largest possible treasure.
• Lighthouse
Marcel (point J) lies in the grass and sees the top of the tent (point T) and behind it the top of the lighthouse (P). | TT '| = 1.2m, | PP '| = 36m, | JT '| = 5m. Marcel lies 15 meters away from the sea (M). Calculate the lighthouse distance from the sea
• Trapezium diagonals
It is given trapezium ABCD with bases | AB | = 12 cm, |CD| = 8 cm. Point S is the intersection of the diagonals for which |AS| is 6 cm long. Calculate the length of the full diagonal AC.
• Similar triangles
The triangles ABC and XYZ are similar. Find the missing lengths of the sides of the triangles. a) a = 5 cm b = 8 cm x = 7.5 cm z = 9 cm b) a = 9 cm c = 12 cm y = 10 cm z = 8 cm c) b = 4 cm c = 8 cm x = 4.5 cm z = 6 cm
• The triangles
The triangles KLM and ABC are given, which are similar to each other. Calculate the lengths of the remaining sides of the triangle KLM, if the lengths of the sides are a = 7 b = 5.6 c = 4.9 k = 5
• Sides of right angled triangle
One leg is 1 m shorter than the hypotenuse, and the second leg is 2 m shorter than the hypotenuse. Find the lengths of all sides of the right-angled triangle.
• Right circular cone
The volume of a right circular cone is 5 liters. Calculate the volume of the two parts into which the cone is divided by a plane parallel to the base, one-third of the way down from the vertex to the base.
• The triangles
The triangles ABC and A'B'C 'are similar with a similarity coefficient of 2. The angles of the triangle ABC are alpha = 35°, beta = 48°. Determine the magnitudes of all angles of triangle A'B'C '.
• An observer
An observer standing west of the tower sees its top at an altitude angle of 45 degrees. After moving 50 meters to the south, he sees its top at an altitude angle of 30 degrees. How tall is the tower?
• Cutting cone
A cone with a base radius of 10 cm and a height of 12 cm is given. At what height above the base should we divide it by a section parallel to the base so that the volumes of the two resulting bodies are the same? Express the result in cm.
• Lookout tower
Calculate the height of a lookout tower forming a shadow of 36 m if at the same time a column 2.5 m high has a shadow of 1.5 m.
• Conical bottle
When a conical bottle rests on its flat base, the water in the bottle is 8 cm from it vertex. When the same conical bottle is turned upside down, the water level is 2 cm from its base. What is the height of the bottle?
• Diagonals at right angle
In the trapezoid ABCD this is given: AB=12cm CD=4cm And diagonals crossed under a right angle. What is the area of this trapezoid ABCD?