Three altitudes

A triangle with altitudes 4; 5, and 6 cm is given. Calculate the lengths of all medians and all sides in a triangle.

Correct result:

a =  7.5236 cm
b =  6.0189 cm
c =  5.0157 cm
t₁ =  4.0671 cm
t₂ =  5.6413 cm
t₃ =  6.3345 cm

Solution:

$$\begin{gathered} v_1=4\text{ cm } \\ {v}_2=5\text{ cm } \\ {v}_3=6\text{ cm } \\ h={\displaystyle \frac{1\mathrm{/}{v}_1+1\mathrm{/}{v}_2+1\mathrm{/}{v}_3}{2}}={\displaystyle \frac{1\mathrm{/}4+1\mathrm{/}5+1\mathrm{/}6}{2}}={\displaystyle \frac{37}{120}}\doteq 0.3083 \\ t=h\cdot (h-1\mathrm{/}{v}_1)\cdot (h-1\mathrm{/}{v}_2)\cdot (h-1\mathrm{/}{v}_3)=0.3083\cdot (0.3083-1\mathrm{/}4)\cdot (0.3083-1\mathrm{/}5)\cdot (0.3083-1\mathrm{/}6)\doteq 0.0003 \\ s=4\cdot \sqrt{t}=4\cdot \sqrt{0.0003}\doteq 0.0665 \\ S={\displaystyle \frac{1}{s}}={\displaystyle \frac{1}{0.0665}}\doteq 15.0472{\text{cm}}^2 \\ S={\displaystyle \frac{a\cdot v_1}{2}} \\ a=2\cdot S\mathrm{/}{v}_1=2\cdot 15.0472\mathrm{/}4=7.5236\text{ cm} \end{gathered}$$

Try calculation via our triangle calculator.

$b={\displaystyle \frac{2\cdot S}{v_2}}={\displaystyle \frac{2\cdot 15.0472}{5}}=6.0189\text{ cm}$

$c={\displaystyle \frac{2\cdot S}{v_3}}={\displaystyle \frac{2\cdot 15.0472}{6}}=5.0157\text{ cm}$

$t_1={\displaystyle \frac{\sqrt{2\cdot b^2+2\cdot c^2-a^2}}{2}}={\displaystyle \frac{\sqrt{2\cdot 6.0189^2+2\cdot 5.0157^2-7.5236^2}}{2}}=4.0671\text{ cm}$

$t_2={\displaystyle \frac{\sqrt{2\cdot a^2+2\cdot c^2-b^2}}{2}}={\displaystyle \frac{\sqrt{2\cdot 7.5236^2+2\cdot 5.0157^2-6.0189^2}}{2}}=5.6413\text{ cm}$

$t_3={\displaystyle \frac{\sqrt{2\cdot a^2+2\cdot b^2-c^2}}{2}}={\displaystyle \frac{\sqrt{2\cdot 7.5236^2+2\cdot 6.0189^2-5.0157^2}}{2}}=6.3345\text{ cm}$

Try another example

Showing 0 comments:

Next similar math problems:

• Medians in right triangle
  It is given a right triangle, angle C is 90 degrees. I know it medians t1 = 8 cm and median t2 = 12 cm. .. How to calculate the length of the sides?
• Medians and sides
  Triangle ABC in the plane Oxy; are the coordinates of the points: A = 2.7 B = -4.3 C-6-1 Try calculate lengths of all medians and all sides.
• Medians in RT
  The rectangular triangle ABC has a length of 10 cm and 24 cm. Points P, Q, R are the centers of the sides of this triangle. The perimeter of the PQR triangle is:
• Rhombus
  The rhombus has diagonal lengths of 4.2cm and 3.4cm. Calculate the length of the sides of the rhombus and its height
• Rectangle 3-4-5
  The sides of the rectangle are in a ratio of 3:4. The length of the rectangle diagonal is 20 cm. Calculate the content of the rectangle.
• Tetrahedral pyramid
  A regular tetrahedral pyramid is given. Base edge length a = 6.5 cm, side edge s = 7.5 cm. Calculate the volume and the area of its face (side area).
• Isosceles trapezoid
  Calculate the content of an isosceles trapezoid whose bases are at ratio 5:3, the arm is 6cm long and it is 4cm high.
• Diamond diagonals
  Calculate the diamond's diagonal lengths if its content is 156 cm² and the side length is 13 cm.
• 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
• Triangle in a square
  In a square ABCD with side a = 6 cm, point E is the center of side AB and point F is the center of side BC. Calculate the size of all angles of the triangle DEF and the lengths of its sides.
• Diagonal
  he rectangular ABCD trapeze, whose AD arm is perpendicular to the AB and CD bases, has area 15cm square. Bases have lengths AB = 6cm, CD = 4cm. Calculate the length of the AC diagonal.
• Ratio of sides
  The triangle has a circumference of 21 cm and the length of its sides is in a ratio of 6: 5: 3. Find the length of the longest side of the triangle in cm.
• Side lengths
  In the triangle ABC, the height to the side a is 6cm. The height to side b is equal to 9 cm. Side "a" is 4 cm longer than side "b". Calculate the side lengths a, b.
• Harmonic mean
  If x, y, z form a harmonic progression, then y is the harmonic mean of x and z. Find the harmonic mean of the numbers 6 and 5.
• Euclid theorems
  Calculate the sides of a right triangle if leg a = 6 cm, and a section of the hypotenuse, which is located adjacent to the second leg b is 5cm.
• Angles and sides of the triangle
  Triangle ABC has a circumference of 26 cm. Lengths of the sides are as follows: a = 11.2 cm; b = 6.5 cm. Arrange the interior angles in order of its size. ?
• Isosceles trapezium
  Calculate the area of an isosceles trapezium ABCD if a = 10cm, b = 5cm, c = 4cm.